On the Classification

of Simple Inductive Limit C^{∗}-Algebras, I:

The Reduction Theorem

Dedicated to Professor Ronald G. Douglas

on the occasion of his sixtieth birthday

Guihua Gong^{1}

Received: October 4, 2002

Communicated by Joachim Cuntz

Abstract. Suppose that

A = lim_{n→∞}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m})

is a simple C^{∗}-algebra, where X_{n,i} are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C^{∗}-algebras is classified by the Elliott invariant — consisting of the ordered K-group and the tracial state space — in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C^{∗}-algebras in this class do not enjoy the real rank zero property.)

1This material is based upon work supported by, or in part by, the U.S. Army Research Office under grant number DAAD19-00-1-0152. The research is also partially supported by NSF grant DMS 9401515, 9622250, 9970840 and 0200739.

Documenta Mathematica 7 (2002) 255–461

Contents

§0 Introduction. §1 Preparation and some preliminary ideas. §2 Spectral multiplicity. §3 Combinatorial results. §4 Decomposition theorems. §5 Almost multiplicative maps. §6 The proof of main theorem.

0 Introduction

In this article and the subsequent article [EGL], we will classify all the unital simple C^{∗}-algebras A, which can be written as the inductive limit of a sequence

⨁_{i=1}^{t_{1}} P_{1,i}M_{[1,i]}(C(X_{1,i}))P_{1,i} ⟶^{ϕ_{1,2}} ⨁_{i=1}^{t_{2}} P_{2,i}M_{[2,i]}(C(X_{2,i}))P_{2,i} ⟶^{ϕ_{2,3}} ⋯,

Simple Inductive Limit C^{∗}-Algebras, I 257

where X_{n,i} are compact metrizable spaces with sup{dim X_{n,i}}_{n,i} < +∞, [n, i] and t_{n} are positive integers, and P_{n,i} ∈ M_{[n,i]}(C(X_{n,i})) are projections. The invariant consists of the ordered K-group and the space of traces on the algebra. The main result in the present article is that a C^{∗}-algebra A as above can be written in another way as an inductive limit so that all the spaces X_{n,i} appearing are certain special simplicial complexes of dimension at most three. Then, in [EGL], the classification theorem will be proved by assuming the C^{∗}- algebras are such special inductive limits. In the special case that the groups K_{∗}(C(X_{n,i})) are torsion free, the C^{∗}-algebra A can be written as an inductive limit of direct sums of matrix algebras over C(S^{1})(i.e., one can replace X_{n,i} by S^{1}). Combining this result with [Ell2], without [EGL]–the part II of this series—, we can still obtain the classification theorem for this special case, which is a generalization of the result of Li for the case that dim(X_{n,i}) = 1 (see [Li1-3]). The theory of C^{∗}-algebras can be regarded as noncommutative topology, and has broad applications in different areas of mathematics and physics (e.g., the study of foliated spaces, manifolds with group actions; see [Con]). One extreme class of C^{∗}-algebras is the class of commutative C^{∗}-algebras, which corresponds to the category of ordinary locally compact Hausdorff topological spaces. The other extreme, which is of great importance, is the class of simple C^{∗}-algebras, which must be considered to be highly noncommutative. For example, the (reduced) foliation C^{∗}-algebra of a foliated space is simple if and only if every leaf is dense in the total space; the cross product C^{∗}-algebra, for a ZZ action on a space X, is simple if and only if the action is minimal. Even though the commutative C^{∗}-algebras and the simple C^{∗}-algebras are opposite extremes, remarkably, many (unital or nonunital) simple C^{∗}-algebras (including the foliation C^{∗}-algebra of a Kronecker foliation, see [EE]) have been proved to be inductive limits of direct sums of matrix algebras over commutative C^{∗}-algebras, i.e., to be of the form lim_{n→∞}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}). (Note that the only commutative C^{∗}- algebras, or matrix algebras over commutative C^{∗}-algebras, which are simple are the very trivial ones, C^{|} or M_{k}(C^{|} ).) In general, it is a conjecture that any stably finite, simple, separable, amenable C^{∗}-algebra is an inductive limit of subalgebras of matrix algebras over commutative C^{∗}-algebras. This conjecture would be analogous to the result of Connes and Haagerup that any amenable von Neumann algebra is generated by an upward directed family of sub von Neumann algebras of type I. The sweeping classification project of G. Elliott is aimed at the complete classification of simple, separable, amenable C^{∗}-algebras in terms of a certain simple invariant, as we mentioned above, consisting of the ordered K-group and the space of traces on the algebra. Naturally, the class of inductive limit

C^{∗}-algebras A = lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}), considered in

this article, is an essential ingredient of the project. Following Blackadar [Bl1], we will call such inductive limit algebras AH algebras. The study of AH algebras has its roots in the theory of AF algebras (see [Br] and [Ell4]). But the modern classification theory of AH algebras was inspired by the seminal paper [Bl3] of B. Blackadar and was initiated by Elliott in [Ell5]. The real rank of a C^{∗}-algebra is the noncommutative counterpart of the dimension of a topological space. Until recently, the only known possibilities for the real rank of a simple C^{∗}-algebra were zero or one. It was proved in [DNNP] that any simple AH algebra

A = lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m})

has real rank either zero or one, provided that sup{dim X_{n,i}}_{n,i} < +∞. (Recently, Villadsen has found a simple C^{∗}-algebra with real rank different from zero and one, see [Vi 2].) For the case of simple C^{∗}-algebras of real rank zero, the classification is quite successful and satisfactory, even though the problem is still not completely solved. Namely, on one hand, the remarkable result of Kirchberg [Kir] and Phillips [Phi1] completely classified all purely infinite, simple, separable, amenable C^{∗}-algebras with the so called UCT property (see also [R] for an important earlier result). All purely infinite simple C^{∗}-algebras are of real rank zero; see [Zh]. On the other hand, in [EG1-2] Elliott and the author completely classified all the stably finite, simple, real rank zero C^{∗}-algebras which are AH algebras of the form lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) with dim(X_{n,i}) ≤ 3. It was proved by Dadarlat and the author that this class includes all simple real rank zero AH algebras with arbitrary but uniformly bounded dimensions for the spaces X_{n,i} (see [D1-2], [G1-4] and [DG]). In this article, the AH algebras considered are not assumed to have real rank zero. As pointed out above, they must have real rank either zero or one. In fact, in a strong sense, almost all of them have real rank one. The real rank zero C^{∗}- algebras are the very special ones for which the space of traces (one part of the invariant mentioned above) is completely determined by the ordered K-group of the C^{∗}-algebra (the other part of the invariant). Not surprisingly, the lack of the real rank zero property presents new essential difficulties. Presumably, dimension one noncommutative spaces are much richer and more complicated than dimension zero noncommutative spaces. In what follows, we would like to explain one of the main differences between the real rank zero case and the general case in the setting of simple AH algebras.

If A = lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}) is of real rank zero, then

Elliott and the author proved a decomposition result (see Theorem 2.21 of [EG2]) which says that ϕ_{n,m} (for m large enough) can be approximately decomposed as a sum of two parts, ϕ_{1} ⊕ ϕ_{2}; one part, ϕ_{1}, having a very small support projection, and the other part, ϕ_{2}, factoring through a finite dimensional algebra. In §4 of the present paper, we will prove a decomposition theorem which says that, for the simple AH algebra A above (with or without the real rank zero condition), ϕ_{n,m} (for m large enough) can be approximately decomposed as a sum of three parts, ϕ_{1} ⊕ ϕ_{2} ⊕ ϕ_{3}: the part ϕ_{1} having a very small support projection compared with the part ϕ_{2}; the part ϕ_{2} factoring through a finite dimensional algebra; and the third part ϕ_{3} factoring through a direct sum of matrix algebras over the interval [0, 1]. (Note that, in the case of a real rank zero inductive limit, the part ϕ_{3} does not appear. In the general case, though, the part ϕ_{3} has a very large support projection compared with the part ϕ_{1}⊕ϕ_{2}.) With this decomposition theorem, we can often deal with the part ϕ_{1} ⊕ ϕ_{2} by using the techniques developed in the classification of the real rank zero case (see [EG1-2], [G1-4], [D1-2] and [DG]). This new decomposition theorem is much deeper. It reflects the real rank one (as opposed to real rank zero) property of the simple C^{∗}-algebra. The special case of the decomposition result that the spaces X_{n,i} are already supposed to be one-dimensional spaces is due to L. Li (see [Li3]). The proof for the case of higher dimensional spaces is essentially more difficult. In particular, as preparation, we need to prove certain combinatorial results (see §3) and also the following result (see §2): Any homomorphism from C(X) to M_{k}(C(Y)) can be perturbed to a homomorphism whose maximum spectral multiplicity (for the definition of this terminology, see 1.2.4 below) is not larger than dim X + dim Y, provided that X ≠ {pt} and X is path connected. The special simplicial complexes used in our main reduction theorem are the following spaces: {pt}, [0, 1], S^{1}, S^{2}, {T_{II,k}}_{k=2}^{∞}, and {T_{III,k}}_{k=2}^{∞}, where the spaces T_{II,k} are two-dimensional connected simplicial complexes with H^{1}(T_{II,k}) = 0 and H^{2}(T_{II,k}) = ZZ/k, and the spaces T_{III,k} are threedimensional connected simplicial complexes with H^{1}(T_{III,k}) = 0 = H^{2}(T_{III,k}) and H^{3}(T_{III,k}) = ZZ/k. (See 4.2 of [EG2] for details.) The spaces T_{II,k} and T_{III,k} are needed to produce the torsion part of the K-groups of the inductive limit C^{∗}-algebras. Since the algebras C(T_{II,k}), C(T_{III,k}), and C(S^{2}) are not stably generated (see [Lo]), difficulties occur in the construction of homomorphisms from these C^{∗}-algebras, when we prove our main reduction theorem (and the isomorphism theorem in [EGL]). In the case of real rank zero algebras, this difficulty can be avoided by using unsuspended E-theory (see [D1-2] and [G1-4]) combined with a certain uniqueness theorem — Theorem 2.29 of [EG2], which only involves homomorphisms (instead of general completely positive linear maps). Roughly speaking, the trouble is that a completely positive linear ∗-contraction, which is an “almost homomorphism”—a G-δ multiplicative map (see 1.1.2 below for the definition of this concept) for sufficiently large G and sufficiently small δ—, may not be automatically close to a homomorphism. As we mentioned above, after we approximately decompose ϕ_{n,m} as ϕ_{1} ⊕ϕ_{2} ⊕ϕ_{3}, we will deal with the part ϕ_{1} ⊕ϕ_{2}, by using the results and techniques from the real rank zero case, in particular by using Theorem 1.6.9 below—a strengthened version of Theorem 2.29 of [EG2]. Therefore, we will consider the composition of the map ϕ_{1} ⊕ ϕ_{2} and a homomorphism from a matrix algebra over {pt}, [0, 1], S^{1}, S^{2}, {T_{II,k}}_{k=2}^{∞}, and {T_{III,k}}_{k=2}^{∞}, to A_{n}. We need this composition to be close to a homomorphism, but ϕ_{1}⊕ϕ_{2} is not supposed to be close to a homomorphism (it is close to the homomorphism ϕ_{n,m} in the case of real rank zero). To overcome the above difficulty, we prove a theorem in §5—a kind of uniqueness theorem, which may be roughly described as follows: For any ε > 0, positive integer N, and finite set F ⊂ A = M_{k}(C(X)), where X is one of the spaces {pt}, [0, 1], T_{II,k}, T_{III,k}, and S^{2}, there are a number δ > 0, a finite set G ⊂ A, and a positive integer L, such that for any two G-δ multiplicative ( see 1.1.2 below), completely positive, linear ∗-contractions ϕ, ψ : M_{k}(C(X)) → B = M_{l}(C(Y)) (where dim(Y) ≤ N), if they define the same map on the level of K-theory and also mod-p K-theory (this statement will be made precise in §5), then there are a homomorphism λ : A → M_{L}(B) with finite dimensional image and a unitary u ∈ M_{L+1}(B) such that

∥(ϕ ⊕ λ)(f) − u(ψ ⊕ λ)(f)u^{∗}∥ < ε

for all f ∈ F. This result is quite nontrivial, and may be expected to have more general applications. Some similar results appear in the literature (e.g., [EGLP, 3.1.4], [D1, Thm A], [G4, 3.9]). But even for ∗-homomorphisms (which are G-δ multiplicative for any G and δ), all these results (except for contractible spaces) require that the number L, the size of the matrix, depends on the maps ϕ and ψ. Note that the theorem stated above does not hold if one replaces X by S^{1}, even if both ϕ and ψ are ∗-homomorphisms. (Fortunately, we do not need the theorem for S^{1} in this article, since C(S^{1}) is stably generated. But on the other hand, the lack of such a theorem for S^{1} causes a major difficulty in the formulation and the proof of the uniqueness theorem involving homomorphisms from C(S^{1}) to M_{k}(C(X)), in [EGL]— part 2 of this series.)

With the above theorem, if a G-δ multiplicative, positive, linear ∗-contraction ϕ and a ∗-homomorphism (name it ψ) define the same map on the level of Ktheory and mod-p K-theory, then ϕ⊕λ is close to a ∗-homomorphism (e.g., Adu∘ (ψ ⊕ λ)) for some ∗-homomorphism λ : A → M_{L}(B) with finite dimensional image. In particular, the size L of the ∗-homomorphism λ can be controlled. This is essential for the construction of ∗-homomorphisms from A = M_{k}(C(X)), where X is one of T_{II,k}, T_{III,k}, and S^{2}. In particular, once L is fixed, we can construct the decomposition of ϕ_{n,m} as ϕ_{1} ⊕ ϕ_{2} ⊕ ϕ_{3}, as mentioned above, such that the supporting projection of the part ϕ_{2} is larger than the supporting projection of the part ϕ_{1} by the amplification of L times. Hence we can prove that, the composition of the map ϕ_{1} ⊕ ϕ_{2} and a homomorphism from a matrix algebra over T_{II,k}, T_{III,k}, and S^{2} to A_{n}, is close to a homomorphism (see Theorems 5.32a and 5.32b below for details). The theorem is also true for a general finite CW complex X, provided that K_{1}(C(X)) is a torsion group. (Note that for the space S^{1} (or the spaces {pt}, [0, 1]), we do not need such a theorem, since any G-δ multiplicative, positive, linear ∗-contraction from M_{k}(C(S^{1})) will automatically be close to a ∗-homomorphism if G is sufficiently large and δ is sufficiently small.) The above mentioned theorem and the decomposition theorem both play important roles in the proof of our main reduction theorem, and also in the proof of the isomorphism theorem in [EGL]. The main results of this article and [EGL] were announced in [G1] and in Elliott’s lecture at the International Congress of Mathematicians in Zurich (see [Ell3]). Since then, several classes of simple inductive limit C^{∗}-algebras have been classified (see [EGJS], [JS 1-2], and [Th1]). But all these later results involve only inductive limits of subhomogeneous algebras with 1-dimensional spectra. In particular, the K_{0}-groups have to be torsion free, since it is impossible to produce the torsion in K_{0}-group with one-dimensional spectra alone, even with subhomogeneous building blocks. This article is organized as follows. In §1, we will introduce some notations, collect some known results, prove some preliminary results, and discuss some important preliminary ideas, which will be used in other sections. In particular, in §1.5, we will discuss the general strategy in the proof of the decomposition theorem, of which, the detailed proof will be given in in §2, §3 and §4. In §1.6, we will prove some uniqueness theorem and factorization theorem which are important in the proof of the main theorem. Even though the results in §1.6 are new, most of the methods are modification of known techniques from [EG2], [D2], [G4] and [DG]. In §2, we will prove the result about maximum spectral multiplicities, which will be used in §4 and other papers. In §3, we will prove certain results of a combinatorial nature. In §4, we will combine the results from §2, §3, and the results in [Li2], to prove the decomposition theorem. In §5, we will prove the result mentioned above concerning G-δ multiplicative maps. In §6, we will use §4, §5 and §1.6 to prove our main reduction theorem. Our main result can be generalized from the case of no dimension growth (i.e., X_{n,i} have uniformly bounded dimensions) to the case of very slow dimension growth. Since the proof of this general case is much more tedious and complicated, we will deal with this generalization in [G5], which can be regarded as an appendix to this article. Acknowledgements. The author would like to thank Professors M. Dadarlat, G. Elliott, L. Li, and H. Lin for helpful conversations. The author also like to thank G. Elliott, L. Li and H. Lin for reading the article and making suggestions to improve the readability of the article. In particular, L. Li suggested to the author to make the pictures (e.g., in 3.6, 3.10 and 6.3) to explain the ideas in the proof of some results; G. Elliott suggested to the author to write a subsection §1.5 to explain the general strategy for proving a decomposition theorem.

1 Preparation and some preliminary ideas

We will introduce some conventions, general assumptions, and preliminary results in this section.

1.1 General assumptions on inductive limits

1.1.1. If A and B are two C^{∗}-algebras, we use Map(A, B) to denote the space of all linear, completely positive ∗-contractions from A to B. If both A and B are unital, then Map(A, B)_{1} will denote the subset of Map(A, B) consisting of unital maps. By word “map”, we shall mean linear, completely positive ∗-contraction between C^{∗}-algebras, or else we shall mean continuous map between topological spaces, which one will be clear from the context. By a homomorphism between C^{∗}-algebras, will be meant a ∗- homomorphism. Let Hom(A, B) denote the space of all homomorphisms from A to B. Similarly, if both A and B are unital, let Hom(A, B)_{1} denote the subset of Hom(A, B) consisting of unital homomorphisms.

Definition 1.1.2. Let G ⊂ A be a finite set and δ > 0. We shall say that ϕ ∈ Map(A, B) is G-δ multiplicative if

∥ϕ(ab) − ϕ(a)ϕ(b)∥ < δ

for all a, b ∈ G. Sometimes, we use Map_{G−δ}(A, B) to denote all the G-δ multiplicative maps.

1.1.3. In the notation for an inductive system (A_{n}, ϕ_{n,m}), we understand that ϕ_{n,m} = ϕ_{m−1,m} ∘ ϕ_{m−2,m−1} ⋯ ∘ ϕ_{n,n+1}, where all ϕ_{n,m} : A_{n} → A_{m} are homomorphisms.

We shall assume that, for any summand A_{n}^{i} in the direct sum A_{n} = ⨁_{i=1}^{t_{n}} A_{n}^{i}, necessarily, ϕ_{n,n+1}(1_{A_{n}^{i}}) ≠ 0, since, otherwise, we could simply delete A_{n}^{i} from A_{n} without changing the limit algebra.

1.1.4. If A_{n} = ⨁_{i} A_{n}^{i} and A_{m} = ⨁_{j} A_{m}^{j}, we use ϕ_{n,m}^{i,j} to denote the partial map of ϕ_{n,m} from the i-th block A_{n}^{i} of A_{n} to the j-th block A_{m}^{j} of A_{m}. In this article, we will assume that all inductive limit C^{∗}-algebras are simple. That is, the limit algebra has no nontrivial proper closed two sided ideals. We will also assume that every inductive limit C^{∗}-algebra A = lim_{→}(A_{n}, ϕ_{n,m}) coming into consideration is different both from M_{k}(C^{|} ) (the matrix algebra over C^{|} ), and from K(H) (the algebra of all compact operators). Since A = lim_{→}(A_{n} = ⨁_{i} A_{n}^{i}, ϕ_{n,m}) is simple, by 5.3.2(b) of [DN], we may

assume that ϕ_{n,m}^{i,j}(1_{A_{n}^{i}}) ≠ 0 for any blocks A_{n}^{i} and A_{m}^{j}, where n < m.

1.1.5. To avoid certain counter examples (see [V]) of the main result of this article, we will restrict our attention, in this article, to inductive systems satisfying the following very slow dimension growth condition. This is a strengthened form of the condition of slow dimension growth introduced in [BDR]. If lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}) is a unital inductive limit system, the very slow dimension growth condition is

lim_{n→+∞} max_{i} \{\frac{(dim X_{n,i})^{3}}{rank(P_{n,i})} \} = 0,

where dim(X_{n,i}) denotes the (covering) dimension of X_{n,i}. In this article, we will also study non-unital inductive limit algebras. The above formula must then be slightly modified. The very slow dimension growth condition in the non-unital case is that, for any summand A_{n}^{i} = P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i} of a fixed A_{n},

lim_{m→+∞} max_{i,j} \{\frac{(dim X_{m,j})^{3}}{rankϕ_{n,m}^{i,j}(1_{A_{n}^{i}})}\} = 0,

where ϕ_{n,m}^{i,j} is the partial map of ϕ_{n,m} from A_{n}^{i} to A_{m}^{j}. (For a unital inductive limit, the two conditions above are equivalent. Of course, both conditions are only proposed for the simple case.) If the set {dim X_{n,i}} is bounded, i.e, there is an M such that

dim X_{n,i} ≤ M

for all n and i, then the inductive system automatically satisfies the very slow dimension growth condition, as we already assume that the limit algebra is not M_{k}(C^{|} ) or K(H).

We will prove our main reduction theorem for the case of uniformly bounded dimensions in this article, since it is significantly simpler than the case of very slow dimension growth. The general case will be discussed in [G5]—an appendix of this article. But the decomposition theorem will be proved for the case of very slow dimension growth. It must be noted that, without the above assumption on dimension growth, the main theorem of this article does not hold (see [Vi1]). We shall leave the following question open: can the above condition of very slow dimension growth be replaced by the similar (but weaker) condition of slow dimension growth (see [BDR]), in the main theorem of this article?

1.1.6. By 2.3 of [Bl1], in the inductive limit

A = lim_{→}(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}),

one can always replace the compact metrizable spaces X_{n,i} by finite simplicial complexes. Note that the replacement does not increase the dimensions of the spaces. Therefore, in this article, we will always assume that all the spaces X_{n,i} in a given inductive system are finite simplicial complexes. Also, we will further assume that all X_{n,i} are path connected. Otherwise, we will separate different components into different direct summands. (Note that a finite simplicial complex has at most finitely many path connected components.) By simplicial complex we mean finite simplicial complex or polyhedron; see [St].

1.1.7.

(a) We use the notation #(·) to denote the cardinal number of the set, if the argument is a finite set. Very often, the sets under consideration will be sets with multiplicity, and then we shall also count multiplicity when we use the notation #.

(b) We shall use a^{∼k} to denote \underbrace{a, ⋯, a}_{k copies} . For example,

{a^{∼3}, b^{∼2}} = {a, a, a, b, b}.

(c) int(·) is used to denote the integer part of a real number. We reserve the notation [·] for equivalence classes in possibly different contexts.

(d) For any metric space X, any x_{0} ∈ X and any c > 0, let B_{c}(x_{0}) := {x ∈ X | d(x, x_{0}) < c} denote the open ball with radius c and centre x_{0}.

(e) Suppose that A is a C^{∗}-algebra, B ⊂ A is a subalgebra, F ⊂ A is a (finite) subset and let ε > 0. If for each element f ∈ F, there is an element g ∈ B such that ∥f − g∥ < ε, then we shall say that F is approximately contained in B to within ε, and denote this by F ⊂_{ε} B.

(f) Let X be a compact metric space. For any δ > 0, a finite set {x_{1}, x_{2}, ⋯ x_{n}} is said to be δ-dense in X, if for any x ∈ X, there is x_{i} such that dist(x, x_{i}) < δ.

(g) We shall use • to denote any possible positive integer. To save notation, y, y′, y′′, ⋯ or a_{1}, a_{2}, ⋯ may be used for finite sequences if we do not care how many terms are in the sequence. Similarly, A_{1} ∪A_{2} ∪⋯ or A_{1} ∩A_{2} ∩⋯ may be used for finite union or finite intersection. If there is a danger of confusion with infinite sequence, union, or intersection, we will write them as a_{1}, a_{2}, ⋯, a_{•}, A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{•}, or A_{1} ∩ A_{2} ∩ ⋯ A_{•}.

(h) For A = ⨁_{i=1}^{t} M_{k_{i}}(C(X_{i})), where X_{i} are path connected simplicial complexes, we use the notation r(A) to denote ⨁_{i=1}^{t} M_{k_{i}}(C^{|} ), which could be considered to be a subalgebra of A consisting of t-tuples of constant functions from X_{i} to M_{k_{i}}(C^{|} ) (i = 1, 2, ⋯, t). Fix a base point x_{i}^{0} ∈ X_{i} for each X_{i}, one can define a map r : A → r(A) by

r(f_{1}, f_{2}, ⋯, f_{t}) = (f_{1}(x_{1}^{0}), f_{2}(x_{2}^{0}), ⋯ f_{t}(x_{t}^{0})) ∈ r(A).

(i) For any two projections p, q ∈ A, we use the notation [p] ≤ [q] to denote that p is unitarily equivalent to a sub projection of q. And we use p ∼ q to denote that p is unitarily equivalent to q.

1.2 Spectrum and spectral variation of a homomorphism

1.2.1. Let Y be a compact metrizable space. Let P ∈ M_{k_{1}}(C(Y)) be a projection with rank(P) = k ≤ k_{1}. For each y, there is a unitary u_{y} ∈ M_{k_{1}}(C^{|} ) (depending on y) such that

P (y) = u_{y} ( \begin{matrix}1 \\ & . & . & . \\ & & & & 1 \\ & & & & & 0 \\ & & & & & & . & . & . \\ & & & & & & & & & 0 \end{matrix} ) u_{y}^{∗},

where there are k 1’s on the diagonal. If the unitary u_{y} can be chosen to be continuous in y, then P is called a trivial projection. It is well known that any projection P ∈ M_{k_{1}}(C(Y)) is locally trivial. That is, for any y_{0} ∈ Y, there is an open set U_{y_{0}} ∋ y_{0}, and there is a continuous unitary-valued function

u : U_{y_{0}} → M_{k_{1}}(C^{|} )

such that the above equation holds for u(y) (in place of u_{y}) for any y ∈ U_{y_{0}}. If P is trivial, then P M_{k_{1}}(C(X))P ≅ M_{k}(C(X)).

1.2.2. Let X be a compact metrizable space and ψ : C(X) → P M_{k_{1}}(C(Y))P be a unital homomorphism. For any given point y ∈ Y, there are points x_{1}(y), x_{2}(y), ⋯, x_{k}(y) ∈ X, and a unitary U_{y} ∈ M_{k_{1}}(C^{|} ) such that

ψ(f)(y) = P (y)U_{y} (\begin{matrix}f(x_{1}(y)) \\ & . & . & . \\ & & & & f(x_{k}(y)) \\ & & & & & 0 \\ & & & & & & . & . & . \\ & & & & & & & & & 0 \end{matrix}) U_{y}^{∗}P (y) ∈ P (y)M_{k_{1}}(C^{|} )P (y)

for all f ∈ C(X). Equivalently, there are k rank one orthogonal projections p_{1}, p_{2}, ⋯, p_{k} with ∑_{i=1}^{k} p_{i} = P (y) and x_{1}(y), x_{2}(y), ⋯, x_{k}(y) ∈ X, such that

ψ(f)(y) = ∑_{i=1}^{k} f(x_{i}(y))p_{i}, ∀ f ∈ C(X).

Let us denote the set {x_{1}(y), x_{2}(y), ⋯, x_{k}(y)}, counting multiplicities, by SPψ_{y}. In other words, if a point is repeated in the diagonal of the above matrix, it is included with the same multiplicity in SPψ_{y}. We shall call SPψ_{y} the spectrum of ψ at the point y. Let us define the spectrum of ψ, denoted by SPψ, to be the closed subset

SPψ := \overline{⋃_{y∈Y} SPψ_{y}} ⊆ X.

Alternatively, SPψ is the complement of the spectrum of the kernel of ψ, considered as a closed ideal of C(X). The map ψ can be factored as

C(X) ⟶^{i^{∗}} C(SPψ) ⟶^{ψ_{1}} P M_{k_{1}}(C(Y))P

with ψ_{1} an injective homomorphism, where i denotes the inclusion SPψ ↪ X. Also, if A = P M_{k_{1}}(C(Y))P, then we shall call the space Y the spectrum of the algebra A, and write SPA = Y (= SP(id)).

1.2.3. In 1.2.2, if we group together all the repeated points in {x_{1}(y), x_{2}(y), ⋯, x_{k}(y)}, and sum their corresponding projections, we can write

ψ(f)(y) = ∑_{i=1}^{l} f(λ_{i}(y))P_{i} (l ≤ k),

where {λ_{1}(y), λ_{2}(y), ⋯, λ_{l}(y)} is equal to {x_{1}(y), x_{2}(y), ⋯, x_{k}(y)} as a set, but λ_{i}(y) ≠ λ_{j}(y) if i ≠ j; and each P_{i} is the sum of the projections corresponding to λ_{i}(y). If λ_{i}(y) has multiplicity m (i.e., it appears m times in {x_{1}(y), x_{2}(y), ⋯, x_{k}(y)}), then rank(P_{i}) = m.

Definition 1.2.4. Let ψ, y, and P_{i} be as above. The maximum spectral multiplicity of ψ at the point y is defined to be max_{i}(rankP_{i}). The maximum spectral multiplicity of ψ is defined to be the supremum of the maximum spectral multiplicities of ψ at the various points of Y.

The following result is the main theorem in §2, which says that we can make the homomorphism not to have too large spectral multiplicities, up to a small perturbation.

Theorem 2.1. Let X and Y be connected simplicial complexes and X ≠ {pt}. Let l = dim(X) + dim(Y). For any given finite set G ⊂ C(X), any ε > 0, and any unital homomorphism ϕ : C(X) → P M_{•}(C(Y))P, where P ∈ M_{•}(C(Y)) is a projection, there is a unital homomorphism ϕ′ : C(X) → P M_{•}(C(Y))P such that

(1) ∥ϕ(g) − ϕ′(g)∥ < ε for all g ∈ G;

(2) ϕ′ has maximum spectral multiplicity at most l.

1.2.5. Set P^{k}(X) = \underbrace{X × X × ⋯ × X}_{k} / ∼, where the equivalence relation ∼

is defined by (x_{1}, x_{2}, ⋯, x_{k}) ∼ (x′_{1}, x′_{2}, ⋯, x′_{k}) if there is a permutation σ of {1, 2, ⋯, k} such that x_{i} = x′_{σ(i)}, for each 1 ≤ i ≤ k. A metric d on X can be extended to a metric on P^{k}(X) by

d([x_{1}, x_{2}, ⋯, x_{k}], [x′_{1}, x′_{2}, ⋯, x′_{k}]) = min_{σ} max_{1≤i≤k} d(x_{i}, x′_{σ(i)}),

where σ is taken from the set of all permutations, and [x_{1}, ⋯, x_{k}] denotes the equivalence class in P^{k}(X) of (x_{1}, ⋯, x_{k}).

1.2.6. Let X be a metric space with metric d. Two k-tuples of (possibly repeating) points {x_{1}, x_{2}, ⋯, x_{k}} ⊂ X and {x′_{1}, x′_{2}, ⋯, x′_{k}} ⊂ X are said to be paired within η if there is a permutation σ such that

d(x_{i}, x′_{σ(i)}) < η, i = 1, 2, ⋯, k.

This is equivalent to the following. If one regards (x_{1}, x_{2}, ⋯, x_{k}) and (x′_{1}, x′_{2}, ⋯, x′_{k}) as two points in P^{k}X, then

d([x_{1}, x_{2}, ⋯, x_{k}], [x′_{1}, x′_{2}, ⋯, x′_{k}]) < η.

1.2.7. Let ψ : C(X) → P M_{k_{1}}(C(Y))P be a unital homomorphism as in 1.2.5. Then

ψ^{∗} : y ↦ SPψ_{y}

defines a map Y → P^{k}X, if one regards SPψ_{y} as an element of P^{k}X. This map is continuous. In term of this map and the metric d, let us define the spectral variation of ψ:

SPV(ψ) := diameter of the image of ψ^{∗}.

Definition 1.2.8. We shall call the projection P_{i} in 1.2.3 the spectral projection of ϕ at y with respect to the spectral element λ_{i}(y). If X_{1} ⊂ X is a subset of X, we shall call

∑_{λ_{i}(y)∈X_{1}} P_{i}

the spectral projection of ϕ at y corresponding to the subset X_{1} (or with respect to the subset X_{1}). In general, for an open set U ⊂ X, the spectral projection P (y) of ϕ at y corresponding to U does not depend on y continuously. But the following lemma holds.

Lemma 1.2.9. Let U ⊂ X be an open subset. Let ϕ : C(X) → M_{•}(C(Y)) be a homomorphism. Suppose that W ⊂ Y is an open subset such that

SPϕ_{y} ∩ (\overline{U}\U) = ∅, ∀y ∈ W.

Then the function

y ↦ spectral projection of ϕ at y correponding to U

is a continuous function on W. Furthermore, if W is connected then #(SPϕ_{y} ∩ U) (counting multiplicity) is the same for any y ∈ W, and the map y ↦ SPϕ_{y}∩ U ∈ P^{l}X is a continuous map on W, where l = #(SPϕ_{y} ∩ U).

Proof: Let P (y) denote the spectral projection of ϕ at y corresponding to the open set U. Fix y_{0} ∈ W. Since SPϕ_{y_{0}} is a finite set, there is an open set U_{1} ⊂ \overline{U_{1}} ⊂ U(⊂ X) such that SPϕ_{y_{0}} ∩ U_{1} = SPϕ_{y_{0}} ∩ U(= SPϕ_{y_{0}} ∩ \overline{U}), or in other words, SPϕ_{y_{0}} ⊂ U_{1} ∪ (X\\overline{U}). Considering the open set U_{1} ∪ (X\\overline{U}), by the continuity of the function

y ↦ SPϕ_{y} ∈ P^{k}X,

where k = rank(ϕ(1)), there is an open set W_{1} ∋ y_{0} such that

(1)  SPϕ_{y} ⊂ U_{1} ∪ (X\\overline{U}), ∀y ∈ W_{1}.

Let χ ∈ C(X) be a function satisfying

χ(x) = \{ \begin{matrix}1 & if & x ∈ \overline{U_{1}} \\ 0 & if & x ∈ X\U . \end{matrix}

Then from (1) and the definition of spectral projection it follows that

ϕ(χ)(y) = P (y), ∀y ∈ W_{1}.

In particular, P (y) is continuous at y_{0}. The additional part of the lemma follows from the continuity of P (y) and the connectedness of W.

⊓ ⊔

In the above proof, we used the following fact, a consequence of the continuity of the map y ↦ SPϕ_{y}. We state it separately for our future use.

Lemma 1.2.10. Let X be a finite simplicial complex, X_{1} ⊂ X be a closed subset, and ϕ : C(X) → M_{•}(C(Y)) be a homomorphism. For any y_{0} ∈ Y, if SPϕ_{y_{0}} ∩ X_{1} = ∅, then there is an open set W ∋ y_{0} such that SPϕ_{y} ∩ X_{1} = ∅ for any y ∈ W. Another equivalent statement is the following. Let U ⊂ X be an open subset. For any y_{0} ∈ Y, if SPϕ_{y_{0}} ⊂ U, then there is an open set W ∋ y_{0} such that SPϕ_{y} ⊂ U for any y ∈ W.

1.2.11. In fact the above lemma is a consequence of the following more general principle: If ϕ : C(X) → M_{•}(C^{|} ) is a homomorphism satisfying SPϕ ⊂ U for a certain open set U, then for any homomorphism ψ : C(X) → M_{•}(C^{|} ) which is close enough to ϕ, we have SPψ ⊂ U. We state it as the following lemma.

Lemma 1.2.12. Let F ⊂ C(X) be a finite set of elements which generate C(X) as a C^{∗}-algebra. For any ε > 0, there is a δ > 0 such that if two homomorphisms ϕ, ψ : C(X) → M_{•}(C^{|} ) satisfy

∥ϕ(f) − ψ(f)∥ < δ, ∀f ∈ F,

then SPψ and SPϕ can be paired within ε. In particular, SPψ ⊂ U, where U is the open set defined by U = {x ∈ X | ∃x′ ∈ SPϕ with dist(x, x′) < ε}.

1.2.13. For any C^{∗} algebra A (usually we let A = C(X) or A = P M_{k}(C(X))P), any homomorphism ϕ : A → M_{•}(C(Y)), and any closed subset Y_{1} ⊂ Y, denote by ϕ|_{Y_{1}} the following composition:

A ⟶^{ϕ} M_{•}(C(Y)) \underrightarrow{restriction} M_{•}(C(Y_{1})).

(As usual, for a subset or subalgebra A_{1} ⊂ A, ϕ|_{A_{1}} will be used to denote the restriction of ϕ to A_{1}. We believe that there will be no danger of confusion as the meaning will be clear from the context.) The following trivial fact will be used frequently.

Lemma 1.2.14. Let Y_{1}, Y_{2} ⊂ Y be two closed subsets. If ϕ_{1} : A → M_{k}(C(Y_{1})) and ϕ_{2} : A → M_{k}(C(Y_{2})) are two homomorphisms with ϕ_{1}|_{Y_{1}∩Y_{2}} = ϕ_{2}|_{Y_{1}∩Y_{2}}, then for any a ∈ A, the matrix-valued function y ↦ ϕ(a)(y), where

ϕ(a)(y) = \{ \begin{matrix}ϕ_{1}(a)(y) & if & y ∈ Y_{1} \\ ϕ_{2}(a)(y) & if & y ∈ Y_{2} , \end{matrix}

is a continuous function on Y_{1} ∪ Y_{2} (i.e., it is an element of M_{k}(C(Y_{1} ∪ Y_{2})). Furthermore, a ↦ ϕ(a) defines a homomorphism ϕ : A → M_{k}(C(Y_{1} ∪ Y_{2})).

1.2.15. Let X be a compact connected space and let Q be a projection of rank n in M_{N}(C(X)). The weak variation of a finite set F ⊂ QM_{N}(C(X))Q is defined by

ω(F) = sup_{Π_{1},Π_{2}} inf_{u∈U(n)} max_{a∈F} ∥uΠ_{1}(a)u^{∗} − Π_{2}(a)∥

where Π_{1}, Π_{2} run through the set of irreducible representations of QM_{N}(C(X))Q into M_{n}(C^{|} ). Let X_{i} be compact connected spaces and Q_{i} ∈ M_{n_{i}}(C(X_{i})) be projections. For a finite set F ⊂ ⨁_{i} Q_{i}M_{n_{i}}(C(X_{i}))Q_{i}, define the weak variation ω(F) to be max_{i} ω(π_{i}(F)), where π_{i} : ⨁_{i} Q_{i}M_{n_{i}}(C(X_{i}))Q_{i} → Q_{i}M_{n_{i}}(C(X_{i}))Q_{i} is the natural project map onto the i-th block. The set F is said to be weakly approximately constant to within ε if ω(F) < ε. The other description of this concept can be found in [EG2, 1.4.11] (see also [D2, 1.3]).

1.2.16. Let ϕ : M_{k}(C(X)) → P M_{l}(C(Y))P be a unital homomorphism. Set ϕ(e_{11}) = p, where e_{11} is the canonical matrix unit corresponding to the upper left corner. Set

ϕ_{1} = ϕ|_{e_{11}M_{k}(C(X))e_{11}} : C(X) ⟶ pM_{l}(C(Y))p.

Then P M_{l}(C(Y))P can be identified with pM_{l}(C(Y))p ⊗ M_{k} in such a way that

ϕ = ϕ_{1} ⊗ 1_{k}.

Let us define

SPϕ_{y} := SP(ϕ_{1})_{y},
SPϕ := SPϕ_{1},
SPV(ϕ) := SPV(ϕ_{1}) .

Suppose that X and Y are connected. Let Q be a projection in M_{k}(C(X)) and ϕ : QM_{k}(C(X))Q → P M_{l}(C(Y))P be a unital map. By the Dilation Lemma (2.13 of [EG2]; see Lemma 1.3.1 below), there are an n, a projection P_{1} ∈ M_{n}(C(Y)), and a unital homomorphism

ϕ̃: M_{k}(C(X)) ⟶ P_{1}M_{n}(C(Y))P_{1} such that

ϕ = ϕ̃|_{QM_{k}(C(X))Q}.

(Note that this implies that P is a subprojection of P_{1}.) We define:

SPϕ_{y} := SPϕ̃_{y},
SPϕ := SPϕ̃,
SPV(ϕ) := SPV(ϕ̃) .

(Note that these definitions do not depend on the choice of the dilation ϕ̃.) The following lemma was essentially proved in [EG2, 3.27] (the additional part is [EG 1.4.13]).

Lemma 1.2.17. Let X be a path connected compact metric space. Let p_{0}, p_{1}, p_{2}, ⋯, p_{n} ∈ M_{•}(C(Y)) be mutually orthogonal projections such that rank(p_{i}) ≥ rank(p_{0}), i = 1, 2, ⋯, n. Let \{x_{1}, x_{2}, ⋯, x_{n}\} be a \frac{δ}{2}-dense subset of X. If a homomorphism ϕ : C(X) → M_{•}(C(Y)) is defined by

ϕ(f) = ϕ_{0}(f) ⊕ ∑_{i=1}^{n} f(x_{i})p_{i},

where ϕ_{0} : C(X) → p_{0}M_{•}(C(Y))p_{0} is an arbitrary homomorphism, then SPV(ϕ) < δ. Consequently, if a finite set F ⊂ C(X) satisfies the condition that ∥f(x) − f(x′)∥ < ε, for any f ∈ F, whenever dist(x, x′) < δ, then ϕ(F) is weakly approximately constant to within ε. (For convenience, we will call such a homomorphism ψ : C(X) → M_{•}(C(Y)), defined by ψ(f) = ∑_{i=1}^{n} f(x_{i})p_{i}, a homomorphism defined by point evaluations on the set {x_{1}, x_{2}, ⋯, x_{n}}.)

Proof: For any two points y, y′ ∈ Y, the sets SPϕ_{y} and SPϕ_{y′} have the following subset in common:

{x_{1}^{∼rank(p_{1})}, x_{2}^{∼rank(p_{2})}, ⋯, x_{n}^{∼rank(p_{n})}}.

The remaining parts of SPϕ_{y} and SPϕ_{y′} are SP(ϕ_{0})_{y} and SP(ϕ_{0})_{y′}, respectively, which have at most rank(p_{0}) elements. It is easy to prove the following fact. For any a, b ∈ X, the sets {a, x_{1}, x_{2}, ⋯, x_{n}} and {b, x_{1}, x_{2}, ⋯, x_{n}} can be paired within δ. In fact, by path connectedness of X and \frac{δ}{2}-density of the set \{x_{1}, x_{2}, ⋯, x_{n}\}, one can find a sequence

a, x_{j_{1}}, x_{j_{2}}, ⋯, x_{j_{k}}, b

beginning with a and ending with b such that each pair of consecutive terms has distance smaller than δ. So {a, x_{j_{1}}, ⋯, x_{j_{k−1}}, x_{j_{k}}} can be paired with {x_{j_{1}}, x_{j_{2}}, ⋯, x_{j_{k}}, b} (= {b, x_{j_{1}}, ⋯, x_{j_{k}}}) one by one to within δ. The other parts of the sets are identical, each element can be paired with itself.

Combining the above fact with the condition that rank(p_{i}) ≥ rank(p_{0}) for any i, we know that SPϕ_{y} and SPϕ_{y′} can be paired within δ. That is, SPV(ϕ) < δ. The rest of the lemma is obvious. Namely, for any two points y, y′, ϕ(f)(y) is approximately unitarily equivalent to ϕ(f)(y′) to within ε, by the same unitary for all f ∈ F (see [EG2, 1.4.13]).

⊓ ⊔

1.2.18. In the last part of the above lemma, one does not need ϕ_{0} to be a homomorphism to guarantee ϕ(F) to be weakly approximately constant to within a small number. In fact, the following is true. Suppose that all the notations are as in 1.2.17 except that the maps ϕ_{0} : C(X) → p_{0}M_{•}(C(Y))p_{0} and ϕ : C(X) → M_{•}(C(Y)) are no longer homomorphisms. Suppose that for any y ∈ Y, there is a homomorphism ψ_{y} : C(X) → p_{0}(y)M_{•}(C^{|} )p_{0}(y) such that

∥ϕ_{0}(f)(y) − ψ_{y}(f)∥ < ε, ∀f ∈ F.

Then the set ϕ(F) is weakly approximately constant to within 3ε. One can prove this claim as follows. For any y ∈ Y, define a homomorphism ϕ_{y} → M_{•}(C^{|} ) by ϕ_{y}(f) = ψ_{y}(f) ⊕ ∑_{i=1}^{n} f(x_{i})p_{i}. Then for any two points y, y′ ∈ Y, as same as in Lemma 1.2.17, SP(ϕ_{y}) and SP(ϕ_{y′}) can be paired within δ. Therefore, ϕ_{y}(f) is approximately unitarily equivalent to ϕ_{y′}(f) to within ε, by the same unitary for all f ∈ F. On the other hand,

∥ϕ(f)(y) − ϕ_{y}(f)∥ < ε and ∥ϕ(f)(y′) − ϕ_{y′}(f)∥ < ε, ∀f ∈ F.

Hence ϕ(f)(y) is approximately unitarily equivalent to ϕ(f)(y′) to within 3ε, by the same unitary for all f ∈ F.

1.2.19. Suppose that F ⊂ M_{k}(C(X)) is a finite set and ε > 0. Let F′ ⊂ C(X) be the finite set consisting of all entries of elements in F and ε′ = \frac{ε}{k}, where k is the order of the matrix algebra M_{k}(C(X)). It is well known that, for any k × k matrix a = (a_{ij}) ∈ M_{k}(B) with entries a_{ij} ∈ B, ∥a∥ ≤ k max_{ij} ∥a_{ij}∥. This implies the following two facts. Fact 1. If ϕ_{1}, ψ_{1} ∈ Map(C(X), B) are (complete positive) linear ∗-contraction (as the notation in 1.1.1) which satisfy

∥ϕ_{1}(f) − ψ_{1}(f)∥ < ε′, ∀f ∈ F′,

then ϕ := ϕ_{1} ⊗ id_{k} ∈ Map(M_{k}(C(X)), M_{k}(B)) and ψ := ψ_{1} ⊗ id_{k} ∈ Map(M_{k}(C(X)), M_{k}(B)) satisfy

∥ϕ(f) − ψ(f)∥ < ε, ∀f ∈ F.

Fact 2. Suppose that ϕ_{1} ∈ Map(C(X), M_{•}(C(Y))) is a (complete positive) linear ∗-contraction. If ϕ_{1}(F′) is weakly approximately constant to within ε′, then ϕ_{1} ⊗ id_{k}(F) is weakly approximately constant to within ε.

Suppose that a homomorphism ϕ_{1} ∈ Hom(C(X), B) has a decomposition described as follows. There exist mutually orthogonal projections p_{1}, p_{2} ∈ B with p_{1} + p_{2} = 1_{B} and ψ_{1} ∈ Hom(C(X), p_{2}Bp_{2}) such that

∥ϕ_{1}(f) − p_{1}ϕ_{1}(f)p_{1} ⊕ ψ_{1}(f)∥ < ε′, ∀f ∈ F′.

Then there is a decomposition for ϕ := ϕ_{1} ⊗ id_{k}:

∥ϕ(f) − P_{1}ϕ_{1}(f)P_{1} ⊕ ψ(f)∥ < ε, ∀f ∈ F,

where ψ := ψ_{1} ⊗ id_{k} and P_{1} = p_{1} ⊗ 1_{k}. In particular, if B = M_{•}(C(Y)) and ψ_{1} is described by

ψ_{1}(f)(y) = ∑ f(α_{i}(y))q_{i}(y), ∀f ∈ C(X),

where ∑ q_{i} = p_{2} and α_{i} : Y → X are continuous maps, then ψ can be described by

ψ(f)(y) = ∑ q_{i}(y) ⊗ f(α_{i}(y)), ∀f ∈ M_{k}(C(X)),

regarding M_{k}(M_{•}(C(Y))) as M_{•}(C(Y)) ⊗ M_{k}. If α_{i} are constant maps, the homomorphism ψ_{1} is called a homomorphism defined by point evaluations as in Lemma 1.2.17. In this case, we will also call the above ψ a homomorphism defined by point evaluations. From the above, we know that to decompose a homomorphism ϕ ∈ Hom(M_{k}(C(X)), M_{•}(C(Y))), one only needs to decompose ϕ_{1} := ϕ|_{e_{11}M_{k}(C(X))e_{11}} ∈ Hom(C(X), ϕ(e_{11})M_{•}(C(Y))ϕ(e_{11})).

1.3 Full matrix algebras, corners, and the dilation lemma

Some results in this article deal with a corner QM_{N}(C(X))Q of the matrix algebra M_{N}(C(X)). But using the following lemma and some other techniques, we can reduce the problems to the case of a full matrix algebra M_{N}(C(X)). The following dilation lemma is Lemma 2.13 of [EG2].

Lemma 1.3.1. (cf. Lemma 2.13 of [EG2]) Let X and Y be any connected finite CW complexes. If ϕ : QM_{k}(C(X))Q → P M_{n}(C(Y))P is a unital homomorphism, then there are an n_{1}, a projection P_{1} ∈ M_{n_{1}}(C(Y)), and a unital homomorphism ϕ̃ : M_{k}(C(X)) → P_{1}M_{n_{1}}(C(Y))P_{1} with the property that QM_{k}(C(X))Q and P M_{n}(C(Y))P can be identified as corner subalgebras of M_{k}(C(X)) and P_{1}M_{n_{1}}(C(Y))P_{1} respectively (i.e., Q and P can be considered to be subprojections of 1_{k} and P_{1}, respectively) and, furthermore, in such a way that ϕ is the restriction of ϕ̃. If ϕ_{t} : QM_{k}(C(X))Q → P M_{n}(C(Y))P, (0 ≤ t ≤ 1) is a path of unital homomorphisms, then there are P_{1}M_{n_{1}}(C(Y))P_{1} (as above) and a path of unital homomorphisms ϕ̃_{t} : M_{k}(C(X)) → P_{1}M_{n_{1}}(C(Y))P_{1} such that QM_{k}(C(X))Q and P M_{n}(C(Y))P are corner subalgebras of M_{k}(C(X)) and P_{1}M_{n_{1}}(C(Y))P_{1} respectively and ϕ_{t} is the restriction of ϕ̃_{t}.

Definition 1.3.2. Let A be a C^{∗}-algebra. A sub-C^{∗}-algebra A_{1} ⊂ A will be called a limit corner subalgebra of A, if there is a sequence of increasing projections

P_{1} ≤ P_{2} ≤ ⋯ ≤ P_{n} ≤ ⋯,

such that A_{1} = \overline{⋃_{n=1}^{∞} P_{n}AP_{n}}. Using Lemma 1.3.1, it is routine to prove the following lemma.

Lemma 1.3.3. (cf. 4.24 of [EG2]) For any AH algebra A = lim_{→}(A_{n} =

⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}), there is an inductive limit Ã = lim_{→}(Ã_{n} =

⨁_{i=1}^{t_{n}} M_{{n,i}}(C(X_{n,i})), ϕ̃_{n,m}) of full matrix algebras over {X_{n,i}}, such that A is isomorphic to a limit corner subalgebra of Ã. In particular, each P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i} is a corner subalgebra of M_{{n,i}}(C(X_{n,i})) and ϕ_{n,m} is the restriction of ϕ̃_{n,m} on A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}. Furthermore, A is stably isomorphic to Ã.

Remark 1.3.4. In the above lemma, in general, the homomorphisms ϕ̃_{n,m} cannot be chosen to be unital, even if all the homomorphisms ϕ_{n,m} are unital. If A is unital, then A can be chosen to be the cut-down of Ã by a single projection (rather than a sequence of projections).

Remark 1.3.5. In Lemma 1.3.3, if A is simple and satisfies the very slow dimension growth condition, then so is Ã. Hence when we consider the nonunital case, we may always assume that A is an inductive limit of direct sums of full matrix algebras over C(X_{n,i}) without loss of generality. Since the reduction theorem in this paper will be proved without the assumption of unitality, we may assume that the C^{∗}-algebra A is an inductive limit of full matrix algebras over finite simplicial complexes. But even in this case, we still need to consider the cut-down P M_{l}(C(X))P of M_{l}(C(X)) in some situations, since the image of a trivial projection may not be trivial. In the proof of the decomposition theorem in §4, we will not assume that A is unital but we will assume that A is the inductive limit of full matrix algebras. Note that a projection in M_{•}(C(X)) corresponds to a complex vector bundle over X. The following result is well known (see Chapter 8 of [Hu]). This result is often useful when we reduce the proof of a result involving the cut-down P M_{l}(C(X))P to the special case of the full matrix algebra M_{l}(C(X)).

Lemma 1.3.6. Let X be a connected simplicial complex and P ∈ M_{l}(C(X)) be a non-zero projection. Let n = rank(P) + dim(X) and m = 2 dim(X) + 1. Then P is Murray-von Neumann equivalent to a subprojection of 1_{n}, and 1_{n} is Murray-von Neumann equivalent to a subprojection of \underbrace{P ⊕ P ⊕ ⋯ P}_{m}, where 1_{n}

is a trivial projection with rank n. Therefore, P M_{l}(C(X))P can be identified as a corner subalgebra of M_{n}(C(X)), and M_{n}(C(X)) can be identified as a corner subalgebra of M_{m}(P M_{l}(C(X))P).

1.4 Topological preliminaries

In this subsection, we will introduce some notations and results in the topology of simplicial complexes. We will also introduce a well known method for the construction of cross sections of a fibre bundle. The content of this subsection may be found in [St], [Hu] and [Wh].

1.4.1. Let X be a connected simplicial complex. Endow X with a metric d as follows. For each n-simplex Δ, one can identify Δ with an n-simplex in IR^{n} whose edges are of length 1 (of course the identification should preserve the affine structure of the simplices). (Such a simplex is the convex hull of n + 1 points {x_{0}, x_{1}, ⋯, x_{n}} in IR^{n} with dist(x_{i}, x_{j}) = 1 for any i ≠ j ∈ {0, 1, ⋯ n}.) Such an identification gives rise to a unique metric on Δ. The restriction of metric d of X to Δ is defined to be the above metric for any simplex Δ ⊂ X. For any two points x, y ∈ X, d(x, y) is defined to be the length of the shortest path connecting x and y. (The length is measured in individual simplexes, by breaking the path into small pieces.) If X is not connected, denote by L the maximum of the diameters of all the connected components. Define d(x, y) = L + 1, if x and y are in different components. (Recall that all the simplicial complexes in this article are supposed to be finite.)

1.4.2. For a simplex Δ, by ∂Δ, we denote the boundary of the simplex Δ, which is the union of all proper faces of Δ. Note that if Δ is a single point— zero dimensional simplex, then ∂Δ = ∅. Obviously, dim(∂Δ) = dim(Δ) − 1. (We use the standard convention that the dimension of the empty space is −1.) By interior(Δ), we denote Δ\∂Δ. Let X be a simplicial complex. Obviously, for each x ∈ X, there is a unique simplex Δ such that x ∈ interior(Δ), which is the simplex Δ of lowest dimension with the condition that x ∈ Δ. (Here we use the fact that if two different simplices of the same dimension intersect, then the intersection is a simplex of lower dimension.) For any simplex Δ, define

Star(Δ) = ⋃{interior(Δ′) | Δ′ ∩ Δ ≠ ∅}.

Then Star(Δ) is an open set which covers Δ. We will use the following two open covers of the simplicial complex X.

(a) For any vertex x ∈ X, let

W_{x} = Star({x})(= ⋃{interior(Δ) | x ∈ Δ}).

Obviously {W_{x}}_{x∈Vertex(X)} is an open cover parameterized by vertices of X.

In this open cover, the intersection W_{x_{1}} ∩ W_{x_{2}} ∩ ⋯ ∩ W_{x_{k}} is nonempty if and only if x_{1}, x_{2}, ⋯, x_{k} span a simplex of X.

(b) We denote the original simplicial structure of X by σ. Introduce a barycentric subdivision (X, τ) of (X, σ). Then for each simplex Δ of (X, σ) (before subdivision), there is exactly one point C_{Δ} ∈ Vertex(X, τ)—the barycenter of Δ, such that C_{Δ} ∈ interior(Δ). (Here interior(Δ) is clearly defined by referring Δ as a simplex of (X, σ).) Define

U_{Δ} = Star_{(X,τ)}({C_{Δ}}).

As in (a), {U_{Δ} | Δ is a simplex of (X, σ)} is an open cover. In fact, U_{Δ} ⊃ interior(Δ). This open cover is parameterized by simplices of (X, σ) (also by vertices of (X, τ), since there is a one to one correspondence between the vertices of (X, τ) and the simplices of (X, σ)). This cover satisfies the following condition: The intersection U_{Δ_{1}} ∩ U_{Δ_{2}} ∩ ⋯ ∩ U_{Δ_{k}} is nonempty if and only if one can reorder the simplices, such that

Δ_{1} ⊂ Δ_{2} ⊂ ⋯ Δ_{n}.

One can verify that the open cover in (b) is a refinement of the open cover in (a).

1.4.3. It is well known that a simplicial complex X is locally contractible. That is, for any point x ∈ X and an open neighborhood U ∋ x, there is an open neighborhood W ∋ x with W ⊂ U such that W can be contracted to a single point inside U. (One can prove this fact directly, using the metric in 1.4.1.) One can endow different metrics on X, but all the metrics are required to induce the same topology as the one in 1.4.1. Using the local contractibility and the compactness of X, one can prove the following fact. For any simplicial complex X with a metric d (may be different from the metric in 1.4.1), there are δ_{X,d} > 0 and a nondecreasing function ρ : (0, δ_{X,d}] → IR^{+} such that the following are true.

(1) lim_{δ→0^{+}} ρ(δ) = 0, and

(2) for any δ ∈ (0, δ_{X,d}] and x_{0} ∈ X, the ball B_{δ}(x_{0}) with radius δ and centre x_{0} (see 1.1.7 (d) for the notation) can be contracted into a single point within the ball B_{ρ(δ)}(x_{0}). I.e., there is a continuous map α : B_{δ}(x_{0}) × [0, 1] → B_{ρ(δ)}(x_{0}) such that

(i) α(x, 0) = x for any x ∈ B_{δ}(x_{0}), (ii) α(x, 1) = x_{0} for any x ∈ B_{δ}(x_{0}). The following lemma is a consequence of the above fact.

Lemma 1.4.4. For any simplicial complex X with metric d, there are δ_{X,d} > 0 and a nondecreasing function ρ : (0, δ_{X,d}] → IR^{+} such that the following are true.

(1) lim_{δ→0^{+}} ρ(δ) = 0, and

(2) for any ball B_{δ}(x_{0}) with radius δ ≤ δ_{X,d}, any simplex Δ (not assumed to be a simplex in X), and any continuous map f : ∂Δ → B_{δ}(x_{0}), there is a continuous map g : Δ → B_{ρ(δ)}(x_{0}) such that g(y) = f(y) for any y ∈ ∂Δ.

Proof: The simplex Δ can be identified with ∂Δ × [0, 1]/∂Δ × {1} in such a way that ∂Δ is identified with ∂Δ × {0}. Define the map g by

g(y, t) = α(f(y), t) ∈ B_{ρ(δ)}(x_{0}), ∀y ∈ ∂Δ, t ∈ [0, 1],

where α is the map in 1.4.3.

⊓ ⊔

1.4.5. The following is a well known result in differential topology: Suppose that M is an m-dimensional smooth manifold, N ⊂ M is an n-dimensional submanifold. If Y is an l-dimensional simplicial complex with l < m − n, then for any continuous map f : Y → M and any ε > 0, there is a continuous map g : Y → M such that

(i) g(Y) ∩ N = ∅ and (ii) dist(g(y), f(y)) < ε, for any y ∈ Y. There is an analogous result in the case of a simplicial complex M and a subcomplex N. Instead of the assumption that M is an m-dimensional smooth manifold, let us suppose that M has the property D(m): for each x ∈ M, there is a contractible open neighborhood U_{x} ∋ x such that U_{x} \{x} is (m−2)- connected, i.e.,

π_{i}(U_{x} \ {x}) = 0 for any i ∈ {0, 1, ⋯ m − 2}.

(We use the following convention: by π_{0}(X) = 0, it will be meant that X is a path connected nonempty space.) Note that IR^{m}\{0} is (m − 2)-connected. Therefore, any m-dimensional manifold has property D(m). The following result is the relative version of Theorem 5.4.16 of [St] (see page 111 of [St]), which also holds according to the top of page 112 of [St].

Proposition 1.4.6. Suppose that M is a simplicial complex with property D(m), and N ⊂ M is a sub-simplicial complex. Suppose that Y is a simplicial complex of dimension l < m − dim(N), and suppose that Y_{1} ⊂ Y is a subsimplicial complex. Suppose that f : Y → M is a continuous map such that f(Y_{1}) ∩ N = ∅. For any ε > 0, there is a continuous map f_{1} : Y → M such that

(i) f_{1}|_{Y_{1}} = f|_{Y_{1}}, (ii) f_{1}(Y) ∩ N = ∅, and (iii) d(f(y), f_{1}(y)) < ε for any y ∈ Y.

1.4.7. Let X, F be two simplicial complexes.

Let Γ ⊂ Homeo(F) be a subgroup of the group of homeomorphisms of the space F. Let us recall the definition of fibre bundle. A fibre bundle over X with fibre F and structure group Γ, is a simplicial complex M with a continuous surjection p : M → X such that the following is true. There is an open cover U of X, and associated to each U ∈ U, there is a homeomorphism

t_{U} : p^{−1}(U) → U × F

(called a local trivialization of the bundle) such that

(1) Each t_{U} takes the fibre of p^{−1}(U) at x ∈ U to the fibre of U × F at the same point x—a trivialization of the restriction p^{−1}(U) of the fibre bundle to U, i.e., the diagram

[figure: p^{−1}(U); t_{U}; U × F; p; p_{1}; U]

is commutative, where p_{1} denotes the project map from the product U × F to the first factor U, and

(2) The given local trivializations differ fibre-wise only by homeomorphisms in the structure group Γ: for any U, V ∈ U and x ∈ U ∩ V,

t_{U} ∘ t_{V}^{−1}|_{{x}×F} ∈ Γ ⊂ Homeo(F).

Furthermore, we will also suppose that the metric d of F is invariant under the action of any element g ∈ Γ, i.e., d(g(x), g(y)) = d(x, y) for any x, y ∈ F. We will see, the fibre bundles constructed in §2, satisfy this condition. (Note that, if F is the vector space IR^{n} with Euclidean metric d, then d is invariant under the action of O(n) ⊂ Homeo(F), but not invariant under the action of Gl(n) ⊂ Homeo(F).) A subset F_{1} ⊂ F is called Γ-invariant if for any g ∈ Γ, g(F_{1}) ⊂ F_{1}.

1.4.8. A cross section of fibre bundle p : M → X is a continuous map f : X → M such that

(p ∘ f)(x) = x for any x ∈ X.

The following Theorem is a consequence of Proposition 1.4.6. The proof is a standard argument often used in the construction of cross sections for fibre bundles (see [Wh]). (In the literature such an argument is often taken for granted.) We give it here for the convenience of the reader.

Theorem 1.4.9. Suppose that p : M → X is a fibre bundle with fibre F. Suppose that F_{1} ⊂ F is a Γ-invariant sub-simplicial complex of F. Suppose that F has the property D(m) and dim(X) < m − dim(F_{1}). Then for any cross section s : X → M and any ε > 0, there is a cross section s_{1} : X → M such that the following two statements are true:

(1) (t_{U} ∘ s_{1})(x) ∉ {x} × F_{1} for any x ∈ U ∈ U, where U is the open cover of X and {t_{U}}_{U∈U} is the local trivialization of the fibre bundle. That is, s_{1}(X) avoids F_{1} in each fibre;

(2) d(s_{1}(x), s(x)) < ε for any x ∈ X, where the distance is taken in the fibre F.

Proof: If the fibre bundle is trivial, then the cross sections of the bundle can be identified with maps from X to F. The conclusion follows immediately from 1.4.6. For the general case, we will use the local trivializations. For each open set U ∈ U, using the trivialization

t_{U} : p^{−1}(U) → U × F,

each cross section f on U induces a continuous map f̃_{U} : U → F by

f̃_{U}(x) = p_{2}(t_{U}(f(x))),

where p_{2} : U × F → F is the projection onto the second factor. Suppose that δ_{F} is as in 1.4.3 (see 1.4.4 also). That is, there is a nondecreasing function ρ : (0, δ_{F}] → (0, ∞) such that lim_{δ→0^{+}} ρ(δ) = 0 and such that any δ-ball B_{δ}(x) can be contracted to a single point within B_{ρ(δ)}(x). Let dim(X) = n. Choose a finite sequence of positive numbers

ε_{n} > ε_{n−1} > ε_{n−2} > ⋯ > ε_{1} > ε_{0} > 0

as follows. Set ε_{n} = min{δ_{F} , ε}. Then choose ε_{n−1} to satisfy

ρ(3ε_{n−1}) < \frac{1}{3}ε_{n} and 3ε_{n−1} < \frac{1}{3}ε_{n}.

Once ε_{l} is defined, then choose ε_{l−1} to satisfy

ρ(3ε_{l−1}) < \frac{1}{3}ε_{l} and 3ε_{l−1} < \frac{1}{3}ε_{l}.

Repeat this procedure until we choose ε_{0} to satisfy

ρ(3ε_{0}) < \frac{1}{3}ε_{1} and 3ε_{0} < \frac{1}{3}ε_{1}.

Let us refine the given simplicial complex structure on X in such a way that each simplex Δ is covered by an open set U ∈ U and that for any simplex Δ and an open set U ∈ U which covers Δ, the map s̃_{U}|_{Δ} : Δ → F, induced by the cross section s, satisfies

diameter(s̃_{U}(Δ)) < ε_{0}.

(Since the metric on F is invariant under the action of any element in Γ, the above inequality holds or not does not depend on the choice of the open set U which covers Δ. In what follows, we will use this fact many times without saying so.) We will apply Proposition 1.4.6 to each simplex of X from the lowest dimension to the highest dimension. For any l ∈ {0, 1, ⋯, n}, let us denote the l-skeleton of X by X^{(l)}. So X^{(n)} = X, and X^{(0)} is the set of vertices of X. Step 1. Fix a vertex x ∈ X^{(0)}, and suppose that x ∈ U ∈ U. Applying Proposition 1.4.6 to {x} (in place of Y with Y_{1} = ∅) and F ( in place of M with N = F_{1}), there exists s̃^{0}(x) ∈ F \ F_{1} such that

d(s̃^{0}(x), s̃_{U}(x)) < ε_{0}.

Any choice of s̃^{0}(x) gives a cross section s^{0} on {x} by

s^{0}(x) = t_{U}^{−1}(x, s̃^{0}(x)),

where (x, s̃^{0}(x)) ∈ {x} × F ⊂ U × F. Defining s^{0} on all vertices, we obtain a cross section s^{0} on X^{(0)} such that

d(s̃_{U}^{0}(x), s̃_{U}(x)) < ε_{0}

for each x ∈ U ∩ X^{(0)}. Step 2. Suppose that for l < n = dim(X), there is a cross section s^{l} : X^{(l)} → M such that for any U ∈ U and any x ∈ X^{(l)} ∩ U, we have s̃_{U}^{l}(x) ∉ F_{1}, and

(∗) d(s̃_{U}^{l}(x), s̃_{U}(x)) < ε_{l}.

Let us define a cross section s^{l+1} : X^{(l+1)} → M as follows. We will work one by one on each (l + 1)-simplex Δ. First, we shall simply extend the cross section s^{l}|_{∂Δ} to a cross section on Δ (see Substep 2.1 below). Then, apply Proposition 1.4.6 to perturb the cross section s^{l}|_{Δ} to avoid F_{1} in each fibre (see Substep 2.2 below). Again, since Proposition 1.4.6 is only for maps (not for cross sections), we will use s̃_{U}^{l}|∂Δ : ∂Δ → F to replace s^{l}|∂Δ, as in Step 1. Substep 2.1. Let Δ be an (l + 1)-simplex. Suppose that Δ ⊂ U ∈ U. Then s̃_{U}^{l}|_{∂Δ} : ∂Δ → F is a continuous map. Since (∗) holds for any x ∈ ∂Δ, and since

diameter(s̃_{U}(Δ)) < ε_{0},

we have

diameter(s̃_{U}^{l}(∂Δ)) < ε_{l} + ε_{l} + ε_{0}.

Let δ = ε_{l} + ε_{l} + ε_{0} < δ_{F}. Then there is a y ∈ F such that s̃_{U}^{l}(∂Δ) ⊂ B_{δ}(y). Since ρ(δ) ≤ ρ(3ε_{l}) < \frac{1}{3}ε_{l+1}, by Lemma 1.4.4, s̃_{U}^{l} : ∂Δ → F can be extended to a map (still denoted by s̃_{U}^{l})

s̃_{U}^{l} : Δ → F,

such that s̃_{U}^{l}(Δ) ⊂ B_{\frac{1}{3} ε_{l+1}}(y). Consequently, the extended map s̃_{U}^{l} also satisfies that diameter(s̃_{U}^{l}(Δ)) < \frac{2}{3}ε_{l+1}. Substep 2.2. Note that s̃_{U}^{l}(x) ∉ F_{1} for any x ∈ ∂Δ. Applying Proposition 1.4.6 to Δ (in place of Y with subcomplex Y_{1} = ∂Δ) and to F (in the place of M with subcomplex N = F_{1}), we obtain a continuous map

s̃^{l+1} : Δ → F

such that

(1) s̃^{l+1}(x) ∉ F_{1} for any x ∈ Δ,

(2) d(s̃^{l+1}(x), s̃_{U}^{l}(x)) < ε_{0}, for any x ∈ Δ, and

(3) s̃^{l+1}|∂Δ = s̃_{U}^{l}|∂Δ. The map s̃^{l+1} defines a cross section s^{l+1} by

s^{l+1}(x) = t_{U}^{−1}(x, s̃^{l+1}(x)).

After working out all the (l + 1)-simplices, we obtain a cross section s^{l+1} on X^{(l+1)}—it is a continuous cross section because it is continuous on each (l +1)- simplex and s^{l+1}|∂Δ = s^{l}|∂Δ from (3) above. Recall that diameter(s̃_{U}(Δ)) < ε_{0} and diameter(s̃_{U}^{l}(Δ)) < \frac{2}{3}ε_{l+1}. Combining these facts with (∗), we have

(∗∗) d(s̃_{U}^{l}(x), s̃_{U}(x)) < ε_{l} + \frac{2}{3}ε_{l+1} + ε_{0}

for any x ∈ Δ. Combining (∗∗) and (2) above, we have

d(s̃_{U}^{l+1}(x), s̃_{U}(x)) < ε_{l} + \frac{2}{3}ε_{l+1} + 2ε_{0} < ε_{l+1}

for any x ∈ X^{l+1} ∩ U. This is (∗) for l + 1 (in place of l). Step 3. By mathematical induction, we can define s^{l} for each l = 0, 1, ⋯ n as the above. Let s_{1} = s^{n} to finish the proof.

⊓ ⊔

The following relative version of the theorem is also true.

Corollary 1.4.10. Suppose that p : M → X is a fibre bundle with fibre F and F_{1} is a Γ-invariant sub-simplicial complex of F. Suppose that F has the property D(m) and that dim(X) < m − dim(F_{1}). Suppose that X_{1} ⊂ X is a sub-simplicial complex. Suppose that the cross section s : X → M satisfies that (t_{U} ∘ s)(x) ∉ {x} × F_{1} for any U ∈ U and any x ∈ X_{1} ∩ U, where U and t_{U} are as in the definition of fibre bundle in 1.4.7. Then for any ε > 0, there is a cross section s_{1} : X → M such that the following three statements are true:

(1) (t_{U} ∘ s_{1})(x) ∉ {x} × F_{1}, for any x ∈ U and U ∈ U;

(2) d(s_{1}(x), s(x)) < ε for any x ∈ X, where the distance is taken inside the fibre F;

(3) s_{1}|_{X_{1}} = s|_{X_{1}}.

Proof: In the proof of Theorem 1.4.9, we have essentially proved this relative version. If fact, in Step 2, we proved that a cross section on a simplex Δ can be constructed within arbitrarily small distance of the original cross section such that

(1) it avoids F_{1} in each fibre, and

(2) it agrees with the original cross section on ∂Δ, provided that the original cross section avoids F_{1} on ∂Δ. This is a local version of the Corollary. To prove the Corollary, one only needs to apply this local version, repeatedly, to the simplices Δ with Δ\∂Δ ⊂ X\X_{1}, from the lowest dimension to the highest dimension.

⊓ ⊔

1.5 About the decomposition theorem

In this subsection, we will briefly discuss the main ideas in the proof of the decomposition theorem stated in §4. Mainly, we will review the ideas in the proofs of special cases already in the literature (see especially [EG2, Theorem 2.21]), point out the additional difficulties in our new setting, and discuss how to overcome these difficulties. This subsection could be skipped without any logical gap, but we do not encourage the reader to do so, except for the expert in the classification theory. By reading this subsection, the reader will get the overall picture of the proof. In particular, how §2, §3, and the results of [Li2] fit into the picture. We will also discuss some ideas in the proof of the combinatorial results of §3. This subsection may also be helpful for understanding the corresponding parts of [EG2], [Li3], and (perhaps) other papers. Even though the discussion in this subsection is sketched, the proof of Lemma 1.5.4 and Propositions 1.5.7 and 1.5.7’ are complete. We will begin our discussion with some very elementary facts.

1.5.1. Let A and B be unital C^{∗}-algebras, and ϕ : A → B, a unital homomorphism. If P ∈ B is a projection which commutes with the image of ϕ, i.e., such that

P ϕ(a) − ϕ(a)P = 0, ∀a ∈ A,

then ϕ(a) can be decomposed into two mutually orthogonal parts ϕ(a)P = P ϕ(a)P and ϕ(a)(1 − P) = (1 − P)ϕ(a)(1 − P):

ϕ(a) = P ϕ(a)P + (1 − P)ϕ(a)(1 − P).

1.5.2. In 1.5.1, let us consider the case that A = C(X). Let F ⊂ C(X) be a finite set. Let unital homomorphism ϕ : C(X) → B and projection P ∈ B be as in 1.5.1. Furthermore, suppose that there is a point x_{0} ∈ X such that P ϕ(f)P = ϕ(f)P is approximately equal to f(x_{0})P to within ε on F:

∥ϕ(f)P − f(x_{0})P ∥ < ε, ∀f ∈ F.

Then

∥ϕ(f) − (1 − P)ϕ(f)(1 − P) ⊕ f(x_{0})P ∥ < ε, ∀f ∈ F.

More generally, if there are mutually orthogonal projections P_{1}, P_{2}, ⋯, P_{n} ∈ B, which commute with ϕ(C(X)), and points x_{1}, x_{2}, ⋯, x_{n} ∈ X such that

(∗) ∥ϕ(f)P_{i} − f(x_{i})P_{i}∥ < ε, ∀f ∈ F, i = 1, 2, ⋯ n,

then

(∗∗) ∥ϕ(f) − (1 − ∑_{i=1}^{n} P_{i})ϕ(f)(1 − ∑_{i=1}^{n} P_{i}) ⊕ ∑_{i=1}^{n} f(x_{i})P_{i}∥ < ε, ∀f ∈ F.

Here, we used the following fact: the norm of the summation of a set of mutually orthogonal elements in a C^{∗}-algebra is the maximum of the norms of all individual elements in the set. In this paper, this fact will be used many times without saying so.

Example 1.5.3. Let F ⊂ C(X) be a finite set, and ε > 0. Choose η > 0 such that if dist(x, x′) < η, then |f(x) − f(x′)| < ε for any f ∈ F. Let x_{1}, x_{2}, ⋯, x_{n} ∈ X be distinct points, and U_{1} ∋ x_{1}, U_{2} ∋ x_{2}, ⋯, U_{n} ∋ x_{n} be mutually disjoint open neighborhoods with U_{i} ⊂ B_{η}(x_{i}) (= {x ∈ X | dist(x, x_{i}) < η}). Consider the case that B = M_{•}(C^{|} ) and let ϕ : C(X) → M_{•}(C^{|} ) be a homomorphism. If P_{i}, i = 1, 2, ⋯, n are the spectral projections corresponding to the open sets U_{i} (see Definition 1.2.4), then the projections P_{i} commute with ϕ(C(X)) and satisfy (*) in 1.5.2. Therefore, the decomposition

(∗∗) ∥ϕ(f) − (1 − ∑_{i=1}^{n} P_{i})ϕ(f)(1 − ∑_{i=1}^{n} P_{i}) ⊕ ∑_{i=1}^{n} f(x_{i})P_{i}∥ < ε,

holds for all f ∈ F. We remark that if #(SPϕ ∩ U_{i}) (counting multiplicities) is large, then, in the decomposition, rank(P_{i})(= #(SPϕ ∩ U_{i})) is large. In the setting of 1.5.2, not only is (**) true for the original projections P_{1}, P_{2}, ⋯, P_{n}, but also it is true for any subprojections p_{1} ≤ P_{1}, p_{2} ≤ P_{2}, ⋯, p_{n} ≤ P_{n}, with ε replaced by 3ε. Namely, the following lemma holds.

Lemma 1.5.4. Let X be a compact metrizable space, and write A = C(X). Let F ⊂ A be a finite set. Let B be a unital C^{∗} -algebra, and ϕ : A → B be a homomorphism. Let ε > 0. Suppose that there are mutually orthogonal projections P_{1}, P_{2}, ⋯, P_{n} in B and points x_{1}, x_{2}, ⋯, x_{n} in X such that P_{i}ϕ(f) = ϕ(f)P_{i} (i = 1, 2, ⋯, n) for any f ∈ C(X) and such that

(∗)  ∥ϕ(f)P_{i} − f(x_{i})P_{i}∥ < ε (i = 1, 2, ⋯, n) for any f ∈ F.

If p_{1}, p_{2}, ⋯, p_{n} are subprojections of P_{1}, P_{2}, ⋯, P_{n} respectively, then

∥ϕ(f) − (1 − ∑_{i=1}^{n} p_{i})ϕ(f)(1 − ∑_{i=1}^{n} p_{i}) ⊕ ∑_{i=1}^{n} f(x_{i})p_{i}∥ < 3ε,

for any f ∈ F. (Notice that the condition that the projections P_{i} commute with ϕ(f) does not by itself imply that the p_{i} almost commute with ϕ(f), but this does follows if (*) holds.) Different versions of this lemma have appeared in a number of papers (especially, [Cu], [GL], [EGLP]).

Proof: The proof is a straightforward calculation. One verifies directly that

∥(∑ P_{i})ϕ(f)(∑ P_{i}) − ∑ f(x_{i})P_{i}∥ < ε, ∀f ∈ F.

Hence on multiplying by 1 − ∑ p_{i} and ∑ p_{i} (one on each side),

∥(1 − ∑ p_{i})ϕ(f)(∑ p_{i})∥ < ε, and ∥(∑ p_{i})ϕ(f)(1 − ∑ p_{i})∥ < ε, ∀f ∈ F;

on multiplying by ∑ p_{i} on both sides,

∥(∑ p_{i})ϕ(f)(∑ p_{i}) − ∑ f(x_{i})p_{i}∥ < ε, ∀f ∈ F.

The desired conclusion follows from identity

ϕ(f) = ((1 − ∑ p_{i}) + ∑ p_{i})ϕ(f)((1 − ∑ p_{i}) + ∑ p_{i}).
⊓ ⊔

Remark 1.5.5. One may wonder why we need the decomposition given in the preceding lemma. In fact, the decomposition (**) of 1.5.2, with the original projections, has a better estimation. Why do we need to use subprojections? The reason is as follows. Suppose that the C^{∗}-algebra A = C(X), the finite set F ⊂ A, the points x_{1}, x_{2}, ⋯, x_{n} ∈ X, and the open sets U_{1} ∋ x_{1}, U_{2} ∋ x_{2}, ⋯, U_{n} ∋ x_{n} are as in 1.5.3. Let us consider the case B = M_{•}(C(Y)) (instead of M_{•}(C^{|} ) in 1.5.3), where Y is a simplicial complex.

Let ϕ : C(X) → M_{•}(C(Y)) be a unital homomorphism. As in 1.5.3, let P_{i}(y) denote the spectral projection of ϕ|_{y} corresponding to the open set U_{i} (see 1.2.8). Then for each y ∈ Y, we have the inequality (**) above,

∥ϕ(f)(y)−(1−∑_{i=1}^{n} P_{i}(y))ϕ(f)(y)(1−∑_{i=1}^{n} P_{i}(y))⊕∑_{i=1}^{n} f(x_{i})P_{i}(y)∥ < ε, ∀f ∈ F.

Unfortunately, P_{i}(y) does not in general depend continuously on y, and so this estimation does not give rise to a decomposition for ϕ globally. On the other hand, one can construct a globally defined continuous projection p_{i}(y) which is a subprojection of P_{i}(y) at each point y, and is such that rank(p_{i}) is not much smaller than min_{y∈Y} rank(P_{i}(y)) (more precisely, rank(p_{i}) ≥ min_{y∈Y} rank(P_{i}(y)) − dim(Y)), by using the continuous selection theorem of [DNNP] as 1.5.6 below. Once this is done , then for each y ∈ Y, applying the lemma, we have

∥ϕ(f)(y)−(1−∑_{i=1}^{n} p_{i}(y))ϕ(f)(y)(1−∑_{i=1}^{n} p_{i}(y))⊕∑_{i=1}^{n} f(x_{i})p_{i}(y)∥ < 3ε, ∀f ∈ F.

Since the projections p_{i}(y) depend continuously on y, they define elements p_{i} ∈ B. We can then rewrite the preceding estimate as

∥ϕ(f) − (1 − ∑_{i=1}^{n} p_{i})ϕ(f)(1 − ∑_{i=1}^{n} p_{i}) ⊕ ∑_{i=1}^{n} f(x_{i})p_{i}∥ < 3ε, ∀f ∈ F.

1.5.6. We would like to discuss how to construct the projections p_{i} referred to in 1.5.5, using the selection theorem [DNNP 3.2]. To guarantee p_{i} to have a large rank, we should assume that P_{i}(y) has a large rank at every point y. So let us assume that for some positive integer k_{i} and for every point y ∈ Y,

#(SPϕ_{y} ∩ U_{i}) ≥ k_{i},

equivalently, rank(P_{i}(y)) ≥ k_{i}. For the sake of simplicity, let us fix i and write U for U_{i} (U ⊂ X), P for P_{i}, k for k_{i}, and p for the desired projection p_{i}. So for every point y ∈ Y,

#(SPϕ_{y} ∩ U) ≥ k,

equivalently, rankP (y) ≥ k. Let us construct a projection p(y), depending continuously on y, such that rankp(y) ≥ k − dim(Y) and p(y) ≤ P (y) for each y ∈ Y. For each fixed y_{0} ∈ Y, since SPϕ_{y_{0}} ∩ U is a finite set, one can choose an open set U′ ⊂ \overline{U′} ⊂ U such that SPϕ_{y_{0}} ∩ U′ = SPϕ_{y_{0}} ∩ U. In particular, SPϕ_{y_{0}} ∩ (\overline{U′}\U′) = ∅. By Lemma 1.2.10, there is a connected open set W ∋ y_{0} in Y such that

SPϕ_{y} ∩ (\overline{U′}\U′) = ∅, ∀y ∈ W.

Let P^{W} (y) be the spectral projection of ϕ_{y} corresponding to open set U′. By Lemma 1.2.9, this depends continuously on y, and so defines a continuous projection-valued function

P^{W} : W → projections of M_{•}(C^{|} ).

Furthermore, P^{W} (y) ≤ P (y) for any y ∈ Y and, for each y in the (connected) subset W,

rank(P^{W} (y)) = #(SPϕ_{y_{0}} ∩ U′) = #(SPϕ_{y_{0}} ∩ U) ≥ k.

Once we have the above locally defined continuous projection-valued functions P^{W} (y), the existence of a globally defined continuous projection-valued function p(y) follows from the following result. Proposition ([DNNP 3.2]). Let Y be a simplicial complex, and let k be a positive integer. Suppose that W is an open covering of Y such that for each W ∈ W, there is a continuous projection-valued map P^{W} : W → M_{•}(C^{|} ) satisfying

rankP^{W} (y) ≥ k for all y ∈ W.

Then there is a continuous projection-valued map p : Y → M_{•}(C^{|} ) such that for each y ∈ Y,

rank p(y) ≥ k − dim(Y), and
p(y) ≤ ⋁{P^{W} (y); W ∈ W, y ∈ W }.

Let p(y) be as given in the preceding proposition with respect to P^{W} as defined above. Then as P^{W} (y) ≤ P (y) for each W, p(y) ≤ P (y) also holds . Recall, we write U for U_{i}, P for P_{i} and p for p_{i}. So, we obtain a projection p_{i} such that p_{i}(y) is a subprojection of P_{i}(y) for every y. Since P_{i}(y), i = 1, 2, ⋯, n, are the spectral projections corresponding to U_{i}, i = 1, 2, ⋯ n, which are mutually disjoint, the projections P_{i}(y), i = 1, 2, ⋯, n, are mutually orthogonal, and so are the projections p_{i}, i = 1, 2, ⋯, n. Combining this construction with Lemma 1.5.4, we have the following result.

Proposition 1.5.7. Let X be a simplicial complex, and F ⊂ C(X) a finite subset. Suppose that ε > 0 and η > 0 are as in 1.5.3, i.e., such that if dist(x, x′) < η, then |f(x) − f(x′)| < ε for any f ∈ F. Suppose that U_{1}, U_{2}, ⋯, U_{n} are disjoint open neighborhoods of (distinct) points x_{1}, x_{2}, ⋯, x_{n} ∈ X, respectively, such that U_{i} ⊂ B_{η}(x_{i}) for all 1 ≤ i ≤ n. Suppose that ϕ : C(X) → M_{•}(C(Y)) is a unital homomorphism, where Y is a simplicial complex, such that

#(SPϕ_{y} ∩ U_{i}) ≥ k_{i} for 1 ≤ i ≤ n, and for all y ∈ Y.

Then there are mutually orthogonal projections p_{1}, p_{2}, ⋯, p_{n} ∈ M_{•}(C(Y)) with rank(p_{i}) ≥ k_{i} − dim(Y) such that

∥ϕ(f) − p_{0}ϕ(f)p_{0} ⊕ ∑_{i=1}^{n} f(x_{i})p_{i}∥ < 3ε for all f ∈ F,

where p_{0} = 1 − ∑ p_{i}. Consequently,

rank(p_{0}) ≤ (#(SPϕ_{y}) − ∑_{i=1}^{n} k_{i}) + n · dim(Y).

(Note that #(SPϕ_{y}) is the order of the matrix algebra M_{•}(C(Y)).) (In fact, the above is also true if one replaces M_{•}(C(Y)) by P M_{•}(C(Y))P, with the exact same proof.)

1.5.8. Proposition 1.5.7 is implicitly contained in the proof of the main decomposition theorem—Theorem 2.21 of [EG2], and explicitly stated as Theorem 2.3 of [Li3], for the case of dimension one. To use 1.5.7 to decompose a partial map ϕ_{m,m′}^{i,j} : M_{[m,i]}(C(X_{m,i})) → M_{[m′,j]}(C(X_{m′,j})) of the connecting homomorphism ϕ_{m,m′} : A_{m} → A_{m′} in the inductive system (A_{m}, ϕ_{m,m′}), we only need to write

ϕ_{m,m′}^{i,j} = ϕ ⊗ id_{[m,i]},

(see 1.2.9), and then decompose ϕ (cf. 1.2.19). In [EG2], we proved that such a map ϕ (for m′ large enough) satisfies the condition in Proposition 1.5.7, if the inductive limit is of real rank zero. More precisely, we constructed mutually disjoint open sets U_{1}, U_{2}, ⋯, U_{n}, with small diameter, such that ∑_{i=1}^{n} k_{i} is very large compared with (#(SPϕ_{y}) − ∑_{i=1}^{n} k_{i}), where k_{i} = min_{y∈Y} #(SPϕ_{y} ∩ U_{i}). (See the open sets W_{i} in the proof of Theorem 2.21 of [EG2].) Therefore, in the above decomposition, the part ∑_{i=1}^{n} f(x_{i})p_{i}, which has rank at least (∑_{i=1}^{n} k_{i}) − n · dim(Y), has much larger size than the size of the part p_{0}ϕ(f)p_{0}, which has rank at most (#(SPϕ_{y}) − ∑_{i=1}^{n} k_{i}) + n · dim(Y), if n · dim(Y) is very small compared with #(SPϕ_{y}). (Notice that if ϕ is not unital, then p_{0} = ϕ(1)−∑_{i=1}^{n} p_{i} and #(SPϕ_{y}) = rankϕ(1).) (We should mention that n·dim(Y) is automatically small from the construction. This is a kind of technical detail, to which the reader should not pay much attention now. The number n depends only on η above, but #(SPϕ_{y}) could be very large as m′ (for ϕ_{m,m′}) is large. In particular, it could be much larger than dim(Y) (note that Y = X_{m′,j} ), if the inductive limit has slow dimension growth.) The above construction is not trivial. It depends heavily on the real rank zero property and Su’s result concerning spectral variation (see [Su]). What was proved by this construction in [EG2] is the decomposition theorem for the real rank zero case, as mentioned in the introduction.

1.5.9. For the case of a non real rank zero inductive system, we can not construct the mutually disjoint open set {U_{i}} as described in 1.5.8. Notice that in the decomposition described in 1.5.8, the major part ψ defined by ψ(f) := ∑_{i=1}^{n} f(x_{i})p_{i} has property that

SPψ_{y} = {x_{1}^{∼rank(p_{1})}, x_{2}^{∼rank(p_{2})}, ⋯, x_{n}^{∼rank(p_{n})}}.

That is, the spectrum consists of several fixed points {x_{i}}_{i=1}^{n}(⊂ X) with multiplicities. This kind of decomposition depends on the real rank zero property. For the decomposition of the simple inductive limit algebras, we are forced to allow the major part to have variable spectrum— SPψ_{y} varies when y varies. The following results can be proved exactly the same as the way Proposition 1.5.7 is proved (see 1.5.6 and 1.5.4). (Proposition 1.5.7 is a special case of the following result by taking α_{i}(y) = x_{i}, the constant maps, and U_{i}(y) = U_{i} ∋ x_{i}, the fixed open sets.)

Proposition 1.5.7’. Let X be a simplicial complex, and F ⊂ C(X), a finite subset. Suppose that ε > 0 and η > 0 are as in 1.5.3, i.e., such that if dist(x, x′) < η, then |f(x) − f(x′)| < ε for any f ∈ F. Suppose that α_{1}, α_{2}, ⋯ α_{n} : Y → X are continuous maps from a simplicial complex Y to X. Suppose that U_{1}(y), U_{2}(y), ⋯, U_{n}(y) are mutually disjoint open sets satisfying U_{i} ⊂ B_{η}(α_{i}(y)) and satisfying the following continuity condition: For any y_{0} ∈ Y and closed set F ⊂ U_{i}(y_{0}), there is an open set W ∋ y_{0} such that F ⊂ U_{i}(y) for any y ∈ W. Suppose that ϕ : C(X) → M_{•}(C(Y)) is a unital homomorphism such that

#(SPϕ_{y} ⋂ U_{i}(y)) ≥ k_{i}, ∀i ∈ {1, 2, ⋯, n}, y ∈ Y.

Then there are mutually orthogonal projections p_{1}, p_{2}, ⋯, p_{n} ∈ M_{•}(C(Y)) with rank(p_{i}) ≥ k_{i} − dim(Y) such that

∥ϕ(f)(y) − p_{0}(y)ϕ(f)(y)p_{0}(y) ⊕ ∑_{i=1}^{n} f(α_{i}(y))p_{i}(y)∥ < 3ε, ∀f ∈ F,

where p_{0} = 1 − ∑ p_{i}. Consequently,

rank(p_{0}) ≤ (#(SPϕ_{y}) − ∑_{i=1}^{n} k_{i}) + n dim(Y).

(It is easy to see that the proof of Proposition 1.5.7 (see 1.5.6) can be generalized to this case. Notice that the above continuity condition for U_{i}(y) assures that \overline{U′} ⊂ U_{i}(y) for any y ∈ W, where U′ is described in 1.5.6 corresponding to y_{0} and U = U_{i}(y_{0}). Then it will assure that P^{W} (y) ≤ P (y), where P^{W} (y) is the locally defined continuous projection-valued function described in 1.5.6.)

1.5.10. In the above proposition, if one can choose the maps α_{i} : Y → X factoring through the interval [0, 1]—the best possible 1-dimensional space—as

α_{i} : Y ⟶^{a_{i}} [0, 1] ⟶^{b} X,

then the part ψ, defined by ψ(f)(y) = ∑_{i=1}^{n} f(α_{i}(y))p_{i}(y), factors through C[0, 1] as

C(X) ⟶^{b^{∗}} C([0, 1]) ⟶^{ψ′} (⨁_{i=1}^{n} p_{i})M_{•}(C(Y))(⨁_{i=1}^{n} p_{i}),

where b^{∗} is induced by b : [0, 1] → X and ψ′ is defined by

ψ′(f)(y) = ∑_{i=1}^{n} f(a_{i}(y))p_{i}(y).

In particular, if

(1)  ∑_{i=1}^{n} k_{i} >> (#(SPϕ_{y}) − ∑_{i=1}^{n} k_{i}),

where k_{i} = min_{y∈Y} #(SPϕ_{y} ∩ U_{i}), then we obtain a decomposition with major part factoring through the interval algebras—direct sum of matrix algebras over interval.

1.5.11. The ideal approach for obtaining a decomposition of ϕ_{m,m′} with major part factoring through an interval algebra, is to reduce it to the setting of Proposition 1.5.7’, that is, to construct continuous maps {α_{i}} (factoring through interval) and the mutually disjoint open sets {U_{i}} as described in 1.5.7’, such that (1) in 1.5.10 holds for homomorphism ϕ induced from the partial connecting homomorphism ϕ_{m,m′}^{i,j} described in 1.5.8. Unfortunately, it seems impossible to realize such a construction globally. A consequence of the property of α_{i} described in Proposition 1.5.7’, is the following property of α_{i}, called property (Pairing): Property (Pairing): For each y ∈ Y, there is a subset of SPϕ_{y}, which can be paired with

{α_{1}(y)^{∼k_{1}}, α_{2}(y)^{∼k_{2}}, ⋯, α_{n}(y)^{∼k_{n}}}

to within η, counting multiplicities. (See 1.1.7 (b) for the notation x^{∼k}.) Even though one can not construct the continuous maps {α_{i}} (factoring through interval [0, 1] and open sets {U_{i}} to satisfy the conditions in Proposition 1.5.7’ together with the condition (1) in 1.5.10, for the connecting homomorphisms in the simple inductive limit, Li constructed the maps {α_{i}} to satisfy the above weaker property (Pairing) and (1) in 1.5.10. In fact, Li proves the following lemma.

Lemma. Suppose that lim(A_{m}, ϕ_{m,m′}) is a simple AH-inductive limit with slow dimension growth and with injective connecting homomorphisms. For any η > 0, and A_{m}, there are a δ > 0 and an integer N > m, such that for any m′ > N, SP(ϕ_{m,m′}^{i,j})_{y} can be paired with

Θ(y) = {α_{1}(y)^{∼T_{1}}, α_{2}(y)^{∼T_{2}}, ⋯, α_{L}(y)^{∼T_{L}}}

to within η, counting multiplicities, for certain continuous maps α_{i} : Y → X factoring through [0,1], where X = X_{m,i} and Y = X_{m′,j}. Furthermore, if

we denote \frac{rank(ϕ_{m,m′}^{i,j} (1_{A^{i}_{m}}))}{rank(1_{A^{i}_{m}})} by K, then T_{i} ≥ int(δK) and therefore L < \frac{2}{δ},

provided δK ≥ 2, where int(δK) is the integer part of δK as in 1.1.7. (This lemma is Theorem 2.19 (see also Remark 2.21) of [Li2]. The additional part about the size of T_{i} could be obtained by inspecting the proof of the theorem (see 2.16 and 2.18 of [Li2]).) (Note that when we write ϕ_{m,m′}^{i,j} = ϕ ⊗ id_{[m,i]}, we have SP(ϕ_{m,m′}^{i,j})_{y} = SPϕ_{y}.) In particular, the condition (1) in 1.5.10 holds, since the right hand side is zero. Therefore the following theorem will be useful for our setting, which is the first theorem in §4.

Theorem 4.1. Let X be a connected finite simplicial complex, and F ⊂ C(X) be a finite set which generates C(X). For any ε > 0, there is an η > 0 such that the following statement is true. Suppose that a unital homomorphism ϕ : C(X) → P M_{•}(C(Y))P (rank(P) = K) (where Y is a finite simplicial complex) satisfies the following condition: There are L continuous maps

a_{1}, a_{2}, ⋯, a_{L} : Y ⟶ X

such that for each y ∈ Y, SPϕ_{y} and Θ(y) can be paired within η, where

Θ(y) = {a_{1}(y)^{∼T_{1}}, a_{2}(y)^{∼T_{2}}, ⋯, a_{L}(y)^{∼T_{L}}}

and T_{1}, T_{2}, ⋯, T_{L} are positive integers with

T_{1} + T_{2} + ⋯ + T_{L} = K = rank(P).

Let T = 2^{L}(dim X + dim Y)^{3}. It follows that there are L mutually orthogonal projections p_{1}, p_{2}, ⋯, p_{L} ∈ P M_{•}(C(Y))P such that

(i) ∥ϕ(f)(y) − p_{0}(y)ϕ(f)(y)p_{0}(y) ⊕ ∑_{i=1}^{L} f(a_{i}(y))p_{i}(y)∥ < ε for any f ∈ F and y ∈ Y, where p_{0} = P − ∑_{i=1}^{L} p_{i}; (ii) ∥p_{0}(y)ϕ(f)(y) − ϕ(f)(y)p_{0}(y)∥ < ε for any f ∈ F and y ∈ Y; (iii) rank(p_{i}) ≥ T_{i} − T for 1 ≤ i ≤ L, and hence rank(p_{0}) ≤ LT. (In the above, η can be chosen to be any number satisfying that if dist(x, x′) < 2η, then |f(x) − f(x′)| < \frac{ε}{3}, ∀f ∈ F.) (Note that we can not make the number T in the above theorem as small as dim(Y), as in Proposition 1.5.7 or 1.5.7’, for some technical difficulties. This is the reason that we are forced to use the stronger condition of very slow dimension growth instead of slow dimension growth in our main decomposition theorem.)

1.5.12. The proof of the above theorem is much more difficult than that of Proposition 1.5.7 or 1.5.7’. In particular, Theorem 2.1 (see §1.2 above), and results in §3 are only for the purpose of proving the above theorem. With these results in hand, the proof of Theorem 4.1 will be given in 4.2–4.19. Then the main decomposition theorem described in the introduction—Theorems 4.35 and 4.37—will be proved based on Theorem 4.1 and the results from [Li2]. We would like to explain the difficulties, and how Theorem 2.1 and §3 will be used to over come the difficulties. Now, our notations are as in Theorem 4.1 above. Fix i. Let U_{i}(y) = {x ∈ X | dist(x, a_{i}(y)) < η}, then from the condition of Theorem 4.1, we have

#(SPϕ_{y} ∩ U_{i}(y)) ≥ T_{i}.

Let P_{i}(y) be the spectral projection corresponding to the open set U_{i}(y). This is not a continuously defined projection. But using the same procedure in 1.5.6 (see 1.5.7 and 1.5.7’), one can construct a globally defined projection p_{i}(y) such that p_{i}(y) ≤ P_{i}(y), and rank(p_{i}(y)) ≥ T_{i} − dim(Y). But unfortunately, those p_{i}(y) are not mutually orthogonal, since U_{i}(y) are not mutually disjoint, and therefore P_{i}(y) are not mutually orthogonal.

1.5.13. If we assume that the maximum spectral multiplicity of ϕ is at most Ω, then for each y ∈ Y, we can divide the set SPϕ_{y} (with multiplicity) into L mutually disjoint subsets E_{1}, E_{2}, ⋯, E_{L} such that, for each λ ∈ E_{i}, dist(λ, a_{i}(y)) ≤ η, i = 1, 2, ⋯, L, and such that

T_{i} − Ω < #(E_{i}) < T_{i} + Ω, i = 1, 2, ⋯, L,

counting multiplicity. By {E_{i}} being mutually disjoint, we mean that if an element λ ∈ SPϕ_{y} has multiplicity k, then we put the entire k copies of λ into one of E_{i}, without separating them. (In the above, ϕ, a_{i}, T_{i}, and L are all from Theorem 4.1.) (Note that if we require that #(E_{i}) = T_{i}, then we can not guarantee {E_{i}} are mutually disjoint, because of spectral multiplicity.) Then we can construct mutually disjoint open sets U_{1}(y), U_{2}(y), ⋯, U_{L}(y) such that E_{i} ⊂ U_{i}(y) and U_{i}(y) ⊂ B_{η}(a_{i}(y)). We can further assume that these open sets have mutually disjoint closures. That is \overline{U_{i}(y)} ∩ \overline{U_{j}(y)} = ∅, for i ≠ j, i, j ∈ {1, 2, ⋯, L}. (The open sets from such construction usually do not satisfy the continuity condition in Proposition 1.5.7’, so we can not apply Proposition 1.5.7’. We need to check the proof of it (e.g. the argument in 1.5.6) against our new setting.)

For each y_{0} ∈ Y, there is an open set W (y_{0}) ∋ y_{0} such that

SPϕ_{y} ⊂ U_{1}(y_{0}) ∪ U_{2}(y_{0}) ∪ ⋯ ∪ U_{L}(y_{0}), ∀y ∈ W (y_{0}).

As in 1.5.6, one can construct the mutually orthogonal locally defined continuous projection-valued functions

P_{i}^{W (y_{0})} : W (y_{0}) → projection of M_{•}(C^{|} ), i = 1, 2, ⋯, L,

where P_{i}^{W (y_{0})}(y) (y ∈ W (y_{0})) are the spectral projections of ϕ_{y} corresponding to open sets U_{i}(y_{0}) (or SPϕ_{y} ∩ U_{i}(y_{0})). Furthermore, rankP_{i}^{W (y_{0})} = #(E_{i}) > T_{i} − Ω. (Note that we do not need to introduce the smaller open set U′ as in 1.5.6, because it is automatically true that SPϕ_{y}∩(\overline{U_{i}(y_{0})}\U_{i}(y_{0})) = ∅, as \{\overline{U_{i}(y_{0})}\}i=1^{L} are mutually disjoint and SPϕ_{y} ⊂ U_{1}(y_{0}) ∪ U_{2}(y_{0}) ∪ ⋯ ∪ U_{L}(y_{0}).) Theorem 2.1 guarantees that Ω is controlled, it is at most dim(X)+ dim(Y), which will be very small compared with T_{i}, in our future application.

1.5.14. There is a finite subcover W = {W (y_{j})}_{j} of the open cover {W (y)}_{y∈Y} of Y. We can use the selection theorem [DNNP 3.2] (see 1.5.6 above) to construct global defined continuous projection valued functions p_{i}(y), i = 1, 2, ⋯, L, of ranks at least T_{i} − Ω − dim(Y), such that

(∗)  p_{i}(y) ≤ ⋁{P_{i}^{W (y_{j})}(y) | y ∈ W (y_{j}) and W (y_{j}) ∈ W}.

For any i_{1} ≠ i_{2} ∈ {1, 2, ⋯, L}, W (y_{j}) ∈ W, and y ∈ W (y_{j}), we have P_{i_{1}}^{W (y_{j})}(y)⊥P_{i_{2}}^{W (y_{j})}(y). Unfortunately, when y ∈ W (y_{j_{1}}) ∩ W (y_{j_{2}}), we do not have

P_{i_{1}}^{W (y_{j_{1}} )}(y)⊥P_{i_{2}}^{W (y_{j_{2}} )}(y).

Therefore, one can not conclude that p_{i_{1}}(y)⊥p_{i_{2}}(y) from the above (*). (Notice that, in the above, P_{i}^{W (y_{0})}(y) is the spectral projection of ϕ_{y} with respect to the open set U_{i}(y_{0}) (not U_{i}(y)), and in general, U_{i_{1}}(y_{j_{1}})∩U_{i_{2}}(y_{j_{2}}) ≠ ∅ if j_{1} ≠ j_{2}. In Propositions 1.5.7 and 1.5.7’, we do not have such problem, since P_{i}^{W} (y) is the spectral projection of U′, which is an open subset of U_{i} (does not depend on y) in the case of Proposition 1.5.7, or which is an open subset of U_{i}(y_{0}) ∩ U_{i}(y) in the case of Proposition 1.5.7’; see 1.5.6 and the explanation after Proposition 1.5.7’ for more details.)

1.5.15. For any W ∈ W and y ∈ W, define Q_{i}^{W} (y), i = 1, 2, ⋯, L, to be the spectral projections of ϕ at point y with respect to the open sets ⋂_{{j: W (y_{j})∩W ≠∅}} U_{i}(y_{j}), i = 1, 2, ⋯, L. These are subprojections of P_{i}^{W} (y).

The advantage of using these projections is the following fact. For any y ∈ W (y_{j_{1}}) ∩ W (y_{j_{2}}), we do have

Q_{i_{1}}^{W (y_{j_{1}} )}(y)⊥Q_{i_{2}}^{W (y_{j_{2}} )}(y)

for any i_{1} ≠ i_{2} ∈ {1, 2, ⋯, L}, because U_{i_{1}}(y_{j_{1}}) ∩ U_{i_{2}}(y_{j_{1}}) = ∅, ⋂_{{j: W (y_{j})∩W (y_{j_{1}} )≠∅}} U_{i_{1}}(y_{j}) ⊂ U_{i_{1}}(y_{j_{1}}) and ⋂_{{j: W (y_{j})∩W (y_{j_{2}} )≠∅}} U_{i_{2}}(y_{j}) ⊂ U_{i_{2}}(y_{j_{1}}) (the second inclusion follows from W (y_{j_{1}}) ∩ W (y_{j_{2}}) ≠ ∅). Then we apply the selection theorem to Q_{i}^{W} (instead of P_{i}^{W}) to find globally defined continuous projection-valued functions p_{i}(y) such that

p_{i}(y) ≤ ⋁{Q_{i}^{W (y_{j})}(y) | y ∈ W (y_{j}) and W (y_{j}) ∈ W},

and such that

rank(p_{i}(y)) ≥ min_{y∈W ∈W}{rank(Q_{i}^{W} (y))} − dim(Y).

(The readers may notice that Q_{i}^{W} (y) are not continuous on W, so one can not apply the selection theorem directly. But one can introduce the open subsets U′ as in 1.5.6 for ⋂_{{j: W (y_{j})∩W ≠∅}} U_{i}(y_{j}) (instead of U_{i}). We omit the details.) This time, p_{i_{1}}(y)⊥p_{i_{2}}(y) for any i_{1} ≠ i_{2} ∈ {1, 2, ⋯, L}. To guarantee rank(p_{i}(y)) to be large—not too much smaller than T_{i}, min_{y∈W ∈W}{rank(Q_{i}^{W} (y))} must be large.

1.5.16. Fixed y_{0} ∈ Y with W (y_{0}) ∈ W. Recall from the definitions of U_{i}(y_{0}) and W (y_{0}) (see 1.5.13),

SPϕ_{y} =(SPϕ_{y} ∩ U_{1}(y_{0}))∪(SPϕ_{y} ∩ U_{2}(y_{0}))∪⋯∪(SPϕ_{y} ∩ U_{L}(y_{0})) , ∀y ∈ W (y_{0}).

Define E_{i}^{W (y_{0})}(y) := SPϕ_{y} ∩ U_{i}(y_{0}). (E_{i}^{W (y_{0})}(y_{0}) is the set E_{i} in 1.5.13, with y_{0} in place of y.) Then for each y ∈ W ∈ W, {E_{i}^{W} (y)}i=1^{L} is a division of SPϕ_{y} (in the terminology in §3, it will be called a grouping of SPϕ_{y}). From 1.5.15,

rank(Q_{i}^{W} (y)) = # (SPϕ_{y} ∩ ⋂_{{j: W (y_{j})∩W ≠∅}} U_{i}(y_{j}))

= #( ⋂_{{j: W (y_{j})∩W ≠∅}} E_{i}^{W (y_{j})}(y)).

Roughly speaking, for our construction in 1.5.15 to work, we need the following condition.

Condition: For each y ∈ Y, the number #(⋂_{{W: y∈W ∈W}} E_{i}^{W} (y)) is large— not too much smaller than T_{i}.

(This condition is a little weaker than Q_{i}^{W} (y) to be large. But we are going to use some special open cover so that the above weaker condition will be enough. We are not going to discuss details here, and the reader does not need to pay much attention.) But in 1.5.15, we only have #(E_{i}^{W} (y)) > T_{i}−Ω. To obtain the above condition, we need the combinatorial results in §3. We are going to discuss it now.

1.5.17. For an intersection to be large, it will be certainly natural to require the number of sets involved in the intersection be as small as possible. As in the setting of 1.5.16, we should require that for any y ∈ Y, the number of sets in W which cover y—#{W | y ∈ W ∈ W}— is not too large. From the definition of covering dimension, we know that for any n-dimensional compact metrizable space Y and any finite cover U of Y, there is a refined cover U_{1} of U such that for any point y ∈ Y, there are at most n + 1 open sets in U_{1} to cover the point y. In particular, for a simplicial complex Y, the construct of such open cover is given in 1.4.2 (a). Let {W_{y}}_{y∈Vertex(Y)} be the open cover of Y given in 1.4.2(a). Recall that, for

any vertices y_{0}, y_{1}, ⋯, y_{k}, the intersection ⋂_{i=1}^{k} W_{y_{i}} is nonempty if and only if y_{0}, y_{1}, ⋯, y_{k} share one simplex. For any finite open cover, there is a finite open cover of the above form (for some refined simplicial complex structure), refining the given open cover. Without loss of generality, we can assume the open cover W = {W (y_{j})} in 1.5.14 is of the above form. Hence {y_{j}} are vertices of the simplicial complex Y and W (y_{j}) are the open sets W_{y_{j}} defined above. Then the condition in 1.5.16 becomes the following. For any simplex Δ of Y with vertices y_{0}, y_{1}, ⋯, y_{k},

# (E_{i}^{W (y_{0})}(y) ∩ E_{i}^{W (y_{1})}(y) ∩ ⋯ ∩ E_{i}^{W (y_{k})}(y)) ≥ T_{i} − C

for any y ∈ W (y_{0}) ∩ W (y_{1}) ∩ ⋯ ∩ W (y_{k}), where C is not too large. (In the proof of Theorem 4.1, the number C will be chosen to be 2^{L}Ω(1 + dim(Y)(dim(Y) + 1)), where Ω is the maximum spectral multiplicity which is bounded by dim(X) + dim(Y), by Theorem 2.1.)

1.5.18. To make the discussion simpler, we suppose that the homomorphism ϕ has distinct spectrum at any point y ∈ Y. That is, the maximum spectral multiplicity of ϕ is one. (Of course, in the proof of Theorem 4.1 in §4, we will not make this assumption.) If the simplicial structure is sufficiently refined, by the distinct property of the spectrum, we can assume the following holds: For any simplex Δ with Z = ⋃_{y∈Vertex(Δ)} W (y)( ⊃ Δ), there are continuous maps

λ_{1}, λ_{2}, ⋯, λ_{K} : Z → X,

where K = rank(P) as in Theorem 4.1, such that

SPϕ_{y} = {λ_{1}(y), λ_{2}(y), ⋯, λ_{K}(y)}, ∀y ∈ Y.

Then for any y ∈ Vertex(Δ), the division (or grouping) E_{i}^{W (y)}(y) of SPϕ_{y} gives rise to a division (or grouping) E_{i}(y) of {λ_{1}, λ_{2}, ⋯, λ_{K}}. In the case of distinct spectrum, the condition in 1.5.13 concerning #(E_{i}) is #(E_{i}(y)) = T_{i}. The condition at the end of 1.5.17 is

# ( ⋂_{y∈Vertex(Δ)} E_{i}(y)) ≥ T_{i} − C.

This is of course not true in general, unless we make some special arrangement. But with the following lemma, we can always subdivide (or refine) the simplicial complex and introduce the groupings of {λ_{1}, λ_{2}, ⋯, λ_{K}} for all newly introduced vertices, to make the above true for any simplex of new simplicial structure (after subdivision). The following formal definition of grouping is in 3.2 of §3. Definition. Let E = {1, 2, ⋯, K} be an index set. Let T_{1}, T_{2}, ⋯, T_{L} be non negative integers with

T_{1} + T_{2} + ⋯ + T_{L} = K.

A grouping of E of type (T_{1}, T_{2}, ⋯, T_{L}) is a set of L mutually disjoint index sets E_{1}, E_{2}, ⋯, E_{L} with

E = E_{1} ∪ E_{2} ∪ ⋯ ∪ E_{L},

and #(E_{j}) = T_{j} for each 1 ≤ j ≤ L. Lemma. Suppose that (Δ, σ) is a simplex, where σ is the standard simplicial structure of the simplex Δ. Suppose that for each vertex x ∈ Vertex(Δ, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{L}(x) of E of type (T_{1}, T_{2}, ⋯, T_{L}). It follows that there is a subdivision (Δ, τ) of (Δ, σ), and there is an extension of the definition of the groupings of E for Vertex(Δ, σ) to the groupings of E (of type (T_{1}, T_{2}, ⋯, T_{L})) for Vertex(Δ, τ) (⊃ Vertex(Δ, σ)) such that:

(1) For each newly introduced vertex x ∈ Vertex(X, τ),

E_{j}(x) ⊂ ⋃_{y∈Vertex(Δ,σ)} E_{j}(y),  j = 1, 2, ⋯, L.

(2) For any simplex Δ_{1} of (X, τ) (after subdivision),

#(  ⋂_{x∈Vertex(Δ_{1})} E_{j}(x)) ≥ T_{j} − \frac{dim(Δ)(dim(Δ) + 1)}{2},  j = 1, 2, ⋯, L.

1.5.19. Condition (1) above is important for the following reason. In 1.5.13, when we define E_{i} as a subset of SPϕ_{y}, we require that dist(λ, a_{i}(y)) < η for any λ ∈ E_{i}. This condition guarantees that the projection P_{i}^{W} in 1.5.13 (or Q_{i}^{W} in 1.5.16) satisfies that ϕ(f)(y)P_{i}^{W} (y) (or ϕ(f)(y)Q_{i}^{W} (y)) is approximately equal to f(a_{i}(y))P_{i}^{W} (y) (or f(a_{i}(y))Q_{i}^{W} (y)) to within ε, which is the condition (*) in Lemma 1.5.4. We consider the grouping of E as the grouping of the spectral functions {λ_{1}, λ_{2}, ⋯, λ_{K}} in 1.5.18. Then the condition (1) in the lemma implies the following fact. If for any vertex y_{0} ∈ Vertex(Δ, σ) and any element k ∈ E_{i}(y_{0}), we have dist(λ_{k}(y), a_{i}(y)) < η, ∀y ∈ Δ, then for any newly introduced vertex y_{1} ∈ Vertex(Δ, τ) and any element k′ ∈ E_{i}(y_{1}) (here E_{i}(y_{1}) is the set E_{i} for the newly introduced grouping for the vertex y_{1}), we still have dist(λ_{k′}(y), a_{i}(y)) < η, ∀y ∈ Δ.

1.5.20. In fact, in the proof of Theorem 4.1, we need the relative version of the result: Suppose that there are a subdivision (∂Δ, τ′) of the boundary (∂Δ, σ) and groupings for all vertices in Vertex(∂Δ, τ′) (⊃ Vertex(Δ, σ)), such that the above (1) holds for any vertex in Vertex(∂Δ, τ′) and such that the above (2) holds for any simplex Δ_{1} of (∂Δ, τ′) with dim(∂Δ) in place of dim(Δ). Then there is a subdivision (Δ, τ) of (Δ, σ), and groupings for all vertices in Vertex(Δ, τ) such that the above (1) and (2) hold and in addition, the following holds: the restriction of (Δ, τ) onto the boundary ∂Δ is (∂Δ, τ′) and the grouping associated to any vertex in Vertex(∂Δ, τ)(= Vertex(∂Δ, τ′)) is the same as the old one. In §3, we will prove the above relative version. In fact, to prove the absolute version will automatically force us to prove the stronger one—the relative version. Another complication comes from the multiplicities, since we can not assume the spectrum to be distinct. In this case, even the definition of grouping needs to be modified.

1.5.21. To give the readers some feeling about the lemma in 1.5.18, we shall discuss the special case of dim(Δ) = 1. That is, Δ = [0, 1], the interval. In this case, we have two groupings for the end points, {E_{1}(0), E_{2}(0), ⋯ E_{L}(0)} and {E_{1}(1), E_{2}(1), ⋯ E_{L}(1)} of E of type (T_{1}, T_{2}, ⋯, T_{L}). Then we need to introduce a sequence of points

0 = t_{0} < t_{1} < t_{2} < ⋯ < t_{n−1} < t_{n} = 1,

(this give rise to a subdivision of Δ = [0, 1]) and define groupings {E_{1}(t_{j}), E_{2}(t_{j}), ⋯ E_{L}(t_{j})} for j = 1, 2, ⋯ n − 1 such that conditions (1) and (2) in the lemma holds. The condition (1) in the lemma means

E_{i}(t_{j}) ⊂ E_{i}(0) ∪ E_{i}(1), ∀i ∈ {1, 2, ⋯ L}, j ∈ {1, 2, ⋯, n − 1}.

The condition (2) in the lemma means

#(E_{i}(t_{j}) ∩ E_{i}(t_{j+1})) ≥ T_{i} − 1,

i.e., for any i, and any pair of adjacent points t_{j}, t_{j+1}, the set E_{i}(t_{j+1}) differs from the set E_{i}(t_{j}) by at most one element.

Let us discuss how to make the above (2) hold for E_{1}. Suppose that E_{1}(0) and E_{1}(1) are different (otherwise, we do not need to do anything for them). We can modify E_{1}(0) to obtain E_{1}(t_{1}) as follows. Taking one element λ in E_{1}(1)\E_{1}(0) to replace an element μ in E_{1}(0)\E_{1}(1), and define it to be E_{1}(t_{1}). So E_{1}(t_{1}) contains λ but not μ. Since λ ∉ E_{1}(0), it must be in some E_{i}(0), i > 1. In the set E_{i}(0), after we take out λ and put it into E_{0}(t_{1}), E_{i} has one element less than it should have, so we can put μ in it, and call it E_{i}(t_{1}). For j ≠ 1 or i, E_{j}(t_{1}) should be the same as E_{j}(0). In such a way, we construct the grouping for t_{1}, which satisfies the above (2) for the pair 0, t_{1}. Furthermore, compare to {E_{i}(0)}_{i}, the new grouping {E_{i}(t_{1})}_{i} is one step closer to the grouping {E_{i}(1)}_{i}. Repeating the above construction (e.g., for t_{1} in place of 0) we can construct E_{i}(t_{2}) and so on. Finally, we will reach the grouping at the other end point 1 ∈ [0, 1]. If one does not require the condition (1), the above is the complete proof of the lemma for the one-dimensional case.

1.5.22. Since we require the condition (1), when we add an element λ ∈ E_{1}(1)\E_{1}(0) into E_{1}(0) to define E_{1}(t_{1}) (as in 1.5.21), we shall carefully choose the element μ ∈ E_{1}(0)\E_{1}(1) to be replaced by λ. In §3, we shall prove the following assertion: For any λ ∈ E_{1}(1)\E_{1}(0), there is μ ∈ E_{1}(0)\E_{1}(1) to satisfy the following condition: let F = (E_{1}(0)\{μ}) ∪ {λ}; the set E\F can be grouped into E′_{2}, E′_{3}, ⋯, E′_{L} (#(E′_{i}) = T_{i}), in such a way that

E_{i} ⊂ E_{i}(0) ∪ E_{i}(1).

(Such an element μ is the element we should choose.) This is Lemma 3.9 with E_{i}(0) ∪ E_{i}(1) in place of H_{i}. With the above ideas in mind, it should not be difficult (hopefully) to read the first part of §3, which does not involve multiplicity. The main step of §3 is contained in the proof of Lemma 3.11.

1.5.23. In the case with multiplicity, there are two possible ways of proceeding.

1. Define a grouping of

E = {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{k}^{∼w_{k}}}

to be a (set theoretical) partition of E as a disjoint union of L sets E = E_{1} ∪ E_{2} ∪ ⋯ E_{L}. Using this definition, we have to allow that, at different vertices, the groupings may be of different types. That is, #(E_{i}) may be different for different vertices. (One can compare with T_{i} −Ω < #(E_{i}) < T_{i} +Ω in 1.5.13.)

2. Define a grouping of

E = {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{k}^{∼w_{k}}}

to be a collection of L subsets E_{1}, E_{2}, ⋯, E_{L} with

E_{j} = {λ_{1}^{∼p_{1}^{j}}, λ_{2}^{∼p_{2}^{j}}, ⋯, λ_{k}^{∼p_{k}^{j}}},

where 0 ≤ p_{i}^{j} ≤ w_{j}, such that

∑_{j=1}^{L} p_{i}^{j} = w_{i}, for each i = 1, 2, ⋯, k.

The grouping is called to be of type (T_{1}, T_{2}, ⋯, T_{L}) if

#(E_{j}) = ∑_{i=1}^{k} p_{i}^{j} = T_{j}, for each j = 1, 2, ⋯, L.

In this way, as will be seen, all the groupings corresponding to vertices of a simplicial complex (as in the proof of Theorem 4.1), may be chosen to be of the same type. But the conclusion (2) in the lemma should be modified. Instead of #(⋂_{x∈Vertex(Δ_{1})} E_{j}(x)), j = 1, 2, ⋯, L to be big, we require

#(⋂_{x∈Vertex(Δ_{1})}E_{j}^{∘}(x)), j = 1, 2, ⋯, L, to be big, where for any set F with

multiplicity, F^{∘} is the subset of F consisting of all such elements λ_{i} that {λ_{i}^{∼w_{i}}} are entirely inside F. (See 3.22 for detailed definition of F^{∘}.) In fact, either approaches can be carried out for our purpose. It turns out that the second approach is shorter and more elegant. Therefore we shall take this approach. Even though, in our approach, it is allowed to separate some set {λ_{i}^{∼w_{i}}} into different groups E_{i} of the grouping, we should still group as many whole sets {λ_{j}^{∼w_{j}}} of the index set {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{n}^{∼w_{n}}} as possible into the same set E_{i} of {E_{1}, E_{2}, ⋯, E_{L}}. Assumption 3.27 and Lemma 3.28 are for this purpose. (One needs to pay special attention to the definition and properties of G_{I} in 3.25.) Except this idea, all other parts of the proof are the same as the case of multiplicity one.

1.5.24. Once we have the combinatorial results in §3, and the explanations in 1.5.1–1.5.19, it will not be hard to understand the proof of Theorem 4.1, though there are some other small techniques, which will be clearly explained in the proof (see 4.2–4.19).

1.5.25. Combining Theorem 4.1 and the result of [Li2] (see the lemma stated in 1.5.11), we can obtain a decomposition ϕ_{1} ⊕ψ of ϕ_{m,m′} (for m′ large enough) such that the major part ψ factors through an interval algebra. But to deal with the part ϕ_{1}, we should add to it, a relatively large (comparing with ϕ_{1}) homomorphism ϕ_{2}, which factors through a finite dimensional C^{∗}- algebra—or which is defined by certain point evaluations (on a δ-dense subset of X_{n,i} for some small number δ). In [Li3], Li deals with this problem by another decomposition, taking such a homomorphism out of the part ψ. (She only proved the one dimensional case.) We take a different approach. Going back to the construction of the maps α_{i} in [Li2] (see the lemma inside 1.5.11 above), we can choose sufficiently many of them to be constant maps (see Lemma 4.33 in §4 below). Therefore ψ automatically has such a part defined by point evaluations. We believe that our approach is easier to understand than Li’s approach, though the spirit is the same. Furthermore, our decomposition is a quantitative version (see Theorem 4.35), and is stronger than Li’s theorem even in the case of one dimensional spaces. (This will be important in [EGL].) We shall not use any result from [Li3]. But we encourage the reader to read the short article [Li2], on which our proof heavily depends.

1.6 Some uniqueness theorems and a factorization theorem

First, this subsection contains some uniqueness theorems. In general, a uniqueness theorem states that, under certain conditions, two maps ϕ, ψ : A → B (homomorphisms or completely positive linear contractions between C^{∗}-algebras A and B) are approximately unitarily equivalent to each other to within a given small number ε on a given finite set F ⊂ A, that is, there is a unitary u ∈ B such that

∥ϕ(f) − uψ(f)u^{∗}∥ < ε, ∀f ∈ F.

This subsection also contains a factorization theorem, which says that, there is a homomorphism (in the class of the so called unital simple embeddings) between matrix algebras over (perhaps higher dimensional) spaces, which must approximately factor through a sum of matrix algebras over the special spaces {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}, by means of almost multiplicative maps. We put these two kinds of results together into one subsection, since the proofs of them have some similarity. Also, in the proof of the factorization theorem, we use some uniqueness theorems of this same subsection. Most of the results are modifications of some results in the literature [EG2], [D1-2], [G4] and [DG] (see [Phi], [GL], [Lin1-2] and [EGLP] also). One of the main theorems (Theorem 1.6.9) is a generalization of Theorem 2.29 of [EG2]— the main uniqueness theorem in the classification of real rank zero AH algebras. The proof given here is shorter than the proof given in [EG2]. Another main theorem—Theorem 1.6.26 (see also Corollary 1.6.29) is a refinement of Lemma 2.2 of [D2] (see also Lemma 3.13 and 3.14 of [G4]). Both Theorem 1.6.9 and Corollary 1.6.29 are important in the proof of our main results in §6. The following well known result (see [Lo]) will be used frequently.

Lemma 1.6.1. Suppose that A = ⨁_{i=1}^{t} M_{k_{i}}(C(X_{i})), where X_{i} = {pt}, [0, 1], or S^{1}. For any finite set F ⊂ A and any number ε > 0, there is a finite set G ⊂ A and there is a number δ > 0 such that if C is a C^{∗}-algebra and ϕ ∈ Map(A, C) is a G-δ multiplicative map, then there is a homomorphism ϕ′ ∈ Hom(A, C) satisfying

∥ϕ(f) − ϕ′(f)∥ < ε, ∀f ∈ F.

The following result is essentially contained in [EG2] (see [EG2, 2.11]) and is stated as Theorem 1.2 of [D1] (see also [G4, 3.2 and 3.8]).

Lemma 1.6.2. ([D1, 1.2]) Let X be a finite simplicial complex. For any finite subset F ⊂ C(X) and any number ε > 0, there are a positive integer L, a unital homomorphism τ : C(X) → M_{L}(C(X)), and a unital homomorphism μ : C(X) → M_{L+1}(C(X)) with finite dimensional image such that

∥diag(f, τ(f)) − μ(f)∥ < ε, ∀f ∈ F.

By the argument in 1.2.19, in the above lemma, the algebra C(X) can be replaced by M_{n}(C(X)).

Lemma 1.6.3. Let X be a finite simplicial complex and A = M_{n}(C(X)). For any finite subset F ⊂ A and any number ε > 0, there are a positive integer L, a unital homomorphism τ : A → M_{L}(A)(= M_{nL}(C(X))), and a unital homomorphism μ : A → M_{L+1}(A) with finite dimensional image such that

∥diag(f, τ(f)) − μ(f)∥ < ε, ∀f ∈ F.

Remark 1.6.4. In general, a unital homomorphism λ : C(X) → B with finite dimensional image is always of the form:

λ(f) = ∑ f(x_{i})p_{i}, ∀f ∈ F,

where {x_{i}} is a finite subset of X, and {p_{i}} ⊂ B is a set of mutually orthogonal projections with ∑ p_{i} = 1_{B}. A homomorphism λ : A = M_{n}(C(X)) → B with finite dimensional image is of the form

λ(f) = ∑ p_{i} ⊗ f(x_{i}), ∀f ∈ M_{n}(C(X))

for a certain identification of λ(1_{A})Bλ(1_{A}) ≅ (λ(e_{11})Bλ(e_{11})) ⊗ M_{n}(C^{|} ), where {p_{i}} is a set of mutually orthogonal projections in λ(e_{11})Bλ(e_{11}). The following Lemma is essentially proved in [D1, Lemma 1.4] (see [G4, Theorem 3.9] also), using the idea from [Phi] and [GL].

Lemma 1.6.5. Let X be a finite simplicial complex and A = M_{n}(C(X)). For a finite set F ⊂ M_{n}(C(X)), a positive number ε > 0 and a positive integer N, there are a finite set G ⊂ M_{n}(C(X)), a positive number δ > 0 and a positive integer L, such that the following is true. For any unital C^{∗}-algebra B, any N + 1 completely positive G-δ multiplicative linear ∗-contraction ϕ_{0}, ϕ_{1}, ⋯ ϕ_{N} ∈ Map_{G−δ}(A, B), there are a homomorphism λ ∈ Hom(A, M_{L}(B)) with finite dimensional image and a unitary u ∈ M_{L+1}(B) such that

∥diag(ϕ_{0}(f), λ(f)) − udiag(ϕ_{N}(f), λ(f))u^{∗}∥ < ε + ω, ∀f ∈ F,

where

ω = max_{f∈F} max_{0≤j≤N−1} ∥ϕ_{j}(f) − ϕ_{j+1}(f)∥.

Proof: If we allow the number L to depend on the maps {ϕ_{j}}, then this is Lemma 1.4 of [D1]. In fact, in the proof of [D1, Lemma 1.4], the author proves this stronger version of the lemma. We will not repeat the entire proof in [D1], instead, we will only repeat the construction of G, δ in [D1] and at the same time choose the number L. Apply Lemma 1.6.3 to F ⊂ A, \frac{ε}{4} > 0 to obtain the integer L_{1}, τ : A → M_{L_{1}}(A) and μ : A → M_{L_{1}+1}(A) as in Lemma 1.6.3. Then D := μ(A) is a finite dimensional C^{∗}-subalgebra of M_{L_{1}+1}(A). By Lemma 1.6.1, there are a finite set F_{1} ⊂ D(⊂ M_{L_{1}+1}(A)) and a positive number δ_{1} > 0 such that if C is any C^{∗}-algebra and ψ ∈ Map(D, C) is any F_{1}-δ_{1} multiplicative map, then there is a homomorphism ψ′ ∈ Hom(D, C) such that

∥ψ′(f) − ψ(f)∥ < \frac{ε}{4}, ∀f ∈ μ(F) (⊂ D).

From 1.2.19, there exist a finite set G ⊂ A and a positive number δ > 0 such that if ϕ ∈ Map(A, B) is G-δ multiplicative, then ϕ ⊗ id_{L_{1}+1} ∈ Map(M_{L_{1}+1}(A), M_{L_{1}+1}(B)) is F_{1}-δ_{1} multiplicative. Let L := N(L_{1} + 1). The proof of [D1, Lemma 1.4] proves that such G, δ and L are as desired. (Notice that the size of the homomorphism η on line 9 of page 122 of [D1] is the number L above.)

⊓ ⊔

In Lemma 1.6.5, if we further assume that F ⊂ A is weakly approximately constant to within ε, then one can replace the homomorphism λ in Lemma 1.6.5 by an arbitrary homomorphism with finite dimensional image of sufficiently large size (with ε replaced by 5ε). One can even use two different homomorphisms (provided that the images of the matrix unit e_{11} under these two different homomorphisms are unitarily equivalent) for ϕ_{0} and ϕ_{N}, i.e., instead of diag(ϕ_{0}, λ) and diag(ϕ_{N}, λ), one can use diag(ϕ_{0}, λ_{1}) and diag(ϕ_{N}, λ_{2}) in the estimation. Namely, we can prove the following result.

Corollary 1.6.6. Let X be a finite simplicial complex and A = M_{n}(C(X)). Suppose that ε > 0 and that a finite set F ⊂ M_{n}(C(X)) is weakly approximately constant to within ε. Suppose that N is a positive integer. Then there are a finite set G ⊂ M_{n}(C(X)), a positive number δ > 0, and a positive integer L such that the following is true. For any unital C^{∗}-algebra B and projection p ∈ B, any N+1 completely positive G-δ multiplicative linear ∗-contractions ϕ_{0}, ϕ_{1}, ⋯, ϕ_{N} ∈ Map_{G−δ}(A, pBp), any λ^{1}, λ^{2} ∈ Hom(A, (1 − p)B(1 − p)) with finite dimensional images and with λ^{1}(e_{11}) ∼ λ^{2}(e_{11}) (see 1.1.7(i)) and [λ^{1}(e_{11})] ≥ L · [p], there is a unitary u ∈ B such that

∥diag(ϕ_{0}(f), λ^{1}(f)) − udiag(ϕ_{N}(f), λ^{2}(f))u^{∗}∥ < 5ε + ω, ∀f ∈ F,

where

ω = max_{f∈F} max_{0≤j≤N−1} ∥ϕ_{j}(f) − ϕ_{j+1}(f)∥.

Proof: Suppose that L and λ : A → M_{L}(pBp) are as in Lemma 1.6.5 for the G-δ multiplicative maps ϕ_{0}, ϕ_{1}, ⋯ ϕ_{N} ∈ Map(A, pBp). From 1.6.4, λ is of the following form

λ(f) = ∑_{i=1}^{s} p_{i} ⊗ f(x_{i}), ∀f ∈ M_{n}(C(X))

for a certain identification of λ(1_{A})Bλ(1_{A}) ≅ (λ(e_{11})Bλ(e_{11})) ⊗ M_{n}(C^{|} ), where p_{i}, i = 1, 2, ⋯, s, are mutually orthogonal projections with ∑_{i=1}^{s} p_{i} = λ(e_{11}) ∈ M_{L}(pBp), and {x_{i}} ⊂ X. Fix a base point x_{0} ∈ X. Since F is weakly approximately constant to within ε, for each i, {f(x_{i})}_{f∈F} is approximately unitarily equivalent to {f(x_{0})}_{f∈F} to within ε, one by one by the same unitary. I.e., for each i, there is a unitary v ∈ M_{n}(C^{|} ) such that ∥vf(x_{i})v^{∗} − f(x_{0})∥ < ε, for any f ∈ F. Define newλ by newλ(f) = ∑_{i=1}^{s} p_{i} ⊗ f(x_{0}) = E ⊗ f(x_{0}), where E := λ(e_{11}) = ∑_{i=1}^{s} p_{i}. Then newλ is approximately unitarily equivalent to the old λ to within ε on F. Therefore, with this newλ, we still have

(1) ∥diag(ϕ_{0}(f), λ(f)) − u_{1}diag(ϕ_{N}(f), λ(f))u_{1}∥ < 3ε + ω, ∀f ∈ F,

for some unitary u_{1} ∈ M_{L+1}(pBp). Since λ^{1}(e_{11}) ∼ λ^{2}(e_{11}), without loss of generality, we can assume that λ^{1}|_{M_{n}(C^{|} )} = λ^{2}|_{M_{n}(C^{|} )}, where M_{n}(C^{|} ) ⊂ M_{n}(C(X)) (= A). In particular, λ^{1}(1_{A}) = λ^{2}(1_{A}) and λ^{1}(e_{11}) = λ^{2}(e_{11}). Denote λ^{1}(e_{11}) by E′. Similar to the case of λ, we can assume that

λ^{1}(f) = ∑_{i=1}^{s_{1}} q_{i}^{1} ⊗ f(x_{i}^{1}), ∀f ∈ M_{n}(C(X)),

λ^{2}(f) = ∑_{i=1}^{s_{2}} q_{i}^{2} ⊗ f(x_{i}^{2}), ∀f ∈ M_{n}(C(X))

for a certain identification of λ^{1}(1_{A})Bλ^{1}(1_{A}) ≅ (E′BE′) ⊗ M_{n}(C^{|} ), where {q_{i}^{1}} and {q_{i}^{2}} are two sets of mutually orthogonal projections with ∑_{i=1}^{s_{1}} q_{i}^{1} = ∑_{i=1}^{s_{2}} q_{i}^{2} = E′ ∈ (1 − p)B(1 − p), and {x_{i}^{1}}, {x_{i}^{2}} ⊂ X. Define λ̃ : A → λ^{1}(1_{A})Bλ^{1}(1_{A}) by

λ̃(f) = E′ ⊗ f(x_{0}), ∀f ∈ F.

Similar to the argument for λ, both λ^{1} and λ^{2} are approximately unitarily equivalent to λ̃ to within ε on F.

Since [E] ≤ L·[p] ≤ [E′] (= [λ^{1}(e_{11})]), there is a sub-projection E_{1} ≤ E′ which is unitarily equivalent to E. Write λ̃ = μ_{1} ⊕ μ_{2}, where μ_{1}(f) = E_{1} ⊗ f(x_{0}) and μ_{2}(f) = (E′ − E_{1}) ⊗ f(x_{0}). Then μ_{1} is unitarily equivalent to λ (strictly speaking, newλ). From (1), we have

∥diag(ϕ_{0}(f), μ_{1}(f)) − u_{2}diag(ϕ_{N}(f), μ_{1}(f))u_{2}^{∗}∥ < 3ε + ω, ∀f ∈ F

for a unitary u_{2} ∈ (p⊕E_{1}⊗1_{n})B(p⊕E_{1}⊗1_{n}). Notice that E_{1}⊗1_{n} ≤ E′⊗1_{n} = λ^{1}(1_{A}) ≤ (1 − p). Therefore,

∥diag(ϕ_{0}(f), λ̃(f)) − u_{3}diag(ϕ_{N}(f), λ̃(f))u_{3}^{∗}∥ < 3ε + ω, ∀f ∈ F,

where u_{3} := u_{2} ⊕ ((E′ − E_{1}) ⊗ 1_{n}) ∈ (p ⊕ (E′ ⊗ 1_{n}))B(p ⊕ (E′ ⊗ 1_{n})). We already know that both λ^{1} and λ^{2} are approximately unitarily equivalent to λ̃ on F to within ε, so we have

∥diag(ϕ_{0}(f), λ^{1}(f)) − udiag(ϕ_{N}(f), λ^{2}(f))u^{∗}∥ < 5ε + ω, ∀f ∈ F,

for a unitary u ∈ B.

⊓ ⊔

Lemma 1.6.7. Suppose that A = M_{k}(C(X)), and F ⊂ A is weakly approximately constant to within ε. Suppose that A_{1} is a C^{∗}-algebra, and two homomorphisms ϕ and ψ ∈ Hom(A, A_{1}) are homotopic to each other. There are a finite set G ⊂ A_{1}, a number δ > 0, and a positive integer L > 0 such that the following is true. If B is a unital C^{∗}-algebra, p ∈ B is a projection, λ_{0} ∈ Map(A_{1}, pBp) is G-δ multiplicative, λ_{1} ∈ Hom(A_{1}, (1 − p)B(1 − p)) is a homomorphism with finite dimensional image satisfying [(λ_{1} ∘ ϕ)(e_{11})] ≥ L · [p], and λ ∈ Map(A_{1}, B) is defined by λ = λ_{0} ⊕ λ_{1}, then there is a unitary u ∈ B such that

∥(λ ∘ ϕ)(f) − u(λ ∘ ψ)(f)u^{∗}∥ < 6ε, ∀f ∈ F.

Proof: Since ϕ is homotopic to ψ. There is a continuous path of homomorphisms ϕ_{t}, 0 ≤ t ≤ 1, such that ϕ_{0} = ϕ and ϕ_{1} = ψ. Choose 0 = t_{0} < t_{1} < ⋯ t_{N−1} < t_{N} = 1 such that

∥ϕ_{t_{j+1}}(f) − ϕ_{t_{j}}(f)∥ < ε, ∀j ∈ {0, 1, ⋯, N − 1} and ∀f ∈ F.

Applying Corollary 1.6.6 to ε, F ⊂ A (which is weakly approximately constant to within ε), and the number N from the above, there are G_{1} ⊂ A and δ > 0 and L as in the Corollary 1.6.6. The set G := ⋃_{j=0}^{N} ϕ_{t_{j}}(G_{1}) ⊂ A_{1}, δ > 0 and number L are as desired.

Suppose that λ_{0}, λ_{1} are the maps satisfying the conditions described in the lemma for G, δ, and L as chosen above. Choosing the sequence of G_{1}-δ multiplicative maps in Corollary 1.6.6 to be λ_{0} ∘ϕ_{t_{0}}(= λ_{0} ∘ϕ), λ_{0} ∘ϕ_{t_{1}}, ⋯, λ_{0} ∘ϕ_{t_{N}} (= λ_{0} ∘ ψ), and the homomorphisms λ^{1} and λ^{2} (with finite dimensional images) to be λ^{1} = λ_{1} ∘ ϕ and λ^{2} = λ_{1} ∘ ψ, and using that λ = λ_{0} ⊕ λ_{1}, we have

∥(λ ∘ ϕ)(f) − u(λ ∘ ψ)(f)u^{∗}∥ < 5ε + ω, ∀f ∈ F,

for a certain unitary u ∈ B, where

ω = max_{f∈F} max_{0≤j≤N−1} ∥(λ_{0} ∘ ϕ_{t_{j+1}})(f) − (λ_{0} ∘ ϕ_{t_{j}})(f)∥ < ε,

since λ_{0} is a contraction—norm decreasing map. So the Lemma follows. (Note that if λ_{0} is G-δ multiplicative, then λ_{0} ∘ ϕ_{t_{j}} is G_{1}-δ multiplicative. Also note that we have the condition that [λ_{1} ∘ ϕ(e_{11})] ≥ L · [p]. Another condition λ_{1} ∘ ϕ(e_{11}) ∼ λ_{1} ∘ ψ(e_{11}) follows from the condition ϕ ∼_{h} ψ.)

⊓ ⊔

The author is indebted to Professor G. Elliott for pointing out the proof of the following result to him.

Lemma 1.6.8. Suppose that C is a unital C^{∗}-algebra, and D ⊂ C is a finite dimensional C^{∗}-subalgebra. For any finite set F ⊂ C and any positive number ε > 0, there are a finite set G ⊂ C and a number δ > 0 such that if B is a unital C^{∗}-algebra, λ ∈ Map(C, B) is G-δ multiplicative, then there is a λ′ ∈ Map(C, B) satisfying the following conditions.

1. λ′|_{D} is a homomorphism.

2. ∥λ′(f) − λ(f)∥ < ε, ∀f ∈ F.

Proof: Without loss of generality, we assume that ∥f∥ ≤ 1 for all f ∈ F. By Kasparov’s version of Stinespring Dilation Theorem, for the completely positive linear ∗-contraction λ : C → B, there is a homomorphism ϕ : C → M(B ⊗ K) such that λ(f) = pϕ(f)p for all f ∈ F, where K is the algebra of all compact operators on an infinite dimensional separable Hilbert space, M(B⊗K) is the multiplier algebra of B ⊗ K, and p = 1_{B} ⊗ e_{11} ∈ B ⊗ K ⊂ M(B ⊗ K). For the above λ and ϕ, it is straight forward to check that, for any fixed element a ∈ C, if ∥λ(a · a^{∗}) − λ(a)λ(a^{∗})∥ < δ, then ∥pϕ(a)(1 − p) · (pϕ(a)(1 − p))^{∗}∥ < δ. Therefore, if we choose the finite set G to satisfy that G = G^{∗}, then the G-δ-multiplicativity of the map λ implies the following property of the dilation ϕ and the cutting down projection p:

(∗) ∥ϕ(a) − (pϕ(a)p + (1 − p)ϕ(a)(1 − p))∥ < 2\sqrt{δ}, ∀a ∈ G,

where 1 is the unit of M(B ⊗ K). By a well known perturbation technique (see [Gli] and [Br]), we have the following: If G contains all matrix units e_{ij} of each block of D and δ is small enough, then the above condition (*) implies that there is a unitary u ∈ M(B ⊗K) with ∥u − 1∥ < \frac{ε}{2}, such that

uϕ(D)u^{∗} ⊂ pM(B ⊗ K)p ⊕ (1 − p)M(B ⊗ K)(1 − p).

(One can obtain the above assertion by applying Lemma III.3.2 of [Da] (or even stronger result of [Ch]) with ϕ(D) and pM(B ⊗ K)p ⊕ (1 − p)M(B ⊗ K)(1 − p) in place of U and B in [Da, III.3.2], respectively.) The map λ′ : C → B, defined by λ′(f) = puϕ(f)u^{∗}p, is as desired.

⊓ ⊔

The following result can be considered to be a generalization of Theorem 2.29 of [EG 2].

Theorem 1.6.9. Suppose that A = ⨁_{i=1}^{s} M_{k_{i}}(C(X_{i})) and F ⊂ A is weakly approximately constant to within ε. Suppose that C is a C^{∗}-algebra, the homomorphisms ϕ and ψ ∈ Hom(A, C) are homotopic to each other. There are a finite set G ⊂ C, a number δ > 0, and a positive integer L > 0 such that the following is true. If B is a unital C^{∗}-algebra, p ∈ B is a projection, λ_{0} ∈ Map(C, pBp) is G-δ multiplicative, λ_{1} ∈ Hom(C, (1 − p)B(1 − p)) is a homomorphism with finite dimensional image satisfying that for each i ∈ {1, 2, ⋯, s}, [(λ_{1} ∘ ϕ)(e_{11}^{i})] ≥ L · [p], where e_{11}^{i} is the matrix unit (of upper left corner ) of the i-th block, M_{k_{i}}(C(X_{i})), of A, then there is a unitary u ∈ B such that

∥(λ ∘ ϕ)(f) − u(λ ∘ ψ)(f)u^{∗}∥ < 8ε, ∀f ∈ F,

where λ ∈ Map(A_{1}, B) is defined by λ = λ_{0} ⊕ λ_{1}.

Proof: Let ϕ_{t} be the homotopy between ϕ and ψ. It is well known that there is a unitary path u_{t} ∈ C such that

ϕ_{t}(1_{A^{i}}) = u_{t}ϕ_{0}(1_{A^{i}})u_{t}^{∗},

for all blocks A^{i} = M_{k_{i}}(C(X_{i})). Therefore, without loss of generality, we assume that ϕ(1_{A^{i}}) = ψ(1_{A^{i}}), and that ϕ|_{A^{i}} is homotopic to ψ|_{A^{i}} within the corner ϕ(1_{A^{i}})Cϕ(1_{A^{i}}). Apply Lemma 1.6.7 to ϕ|_{A^{i}}, ψ|_{A^{i}} and π_{i}(F), where π_{i} is the quotient map from A to A^{i}, to obtain G_{1} (⊂ C), δ_{1} and L as G, δ and L in Lemma 1.6.7. For convenience, without loss of generality, we assume that ∥g∥ ≤ 1 for all g ∈ G_{1}. Let E_{i} = ϕ(1_{A^{i}}). Consider the finite dimensional subalgebra D := C^{|} · E_{1} ⊕ C^{|} · E_{2} ⊕ ⋯ ⊕ C^{|} · E_{s} ⊂ C. Applying Lemma 1.6.8, there are G ⊂ C with G ⊃ G_{1} and δ > 0 with δ < \frac{δ_{1}}{3} such that if λ_{0} ∈ Map(C, pBp) is G-δ multiplicative, then there is another map λ′_{0} ∈ Map(C, pBp) satisfying the following conditions.

1. The restriction λ′_{0}|_{D} is a homomorphism.

2. ∥λ′_{0}(f) − λ_{0}(f)∥ < min(\frac{δ_{1}}{3} , ε), ∀f ∈ G_{1} ∪ ϕ(F) ∪ ψ(F).

As a consequence we also have

3. λ′_{0} is G_{1}-δ_{1} multiplicative. The condition 1 above yields that {λ′_{0}(E_{i})}_{i=1}^{s} are mutually orthogonal projections. Such G, δ and L are as desired. Suppose that λ_{0} and λ′_{0} are as above. Set λ′ = λ′_{0} ⊕ λ_{1}. From Lemma 1.6.7 and the ways G_{1}, δ_{1} and L are chosen, there are unitaries u_{i} ∈ λ′(E_{i})Bλ′(E_{i}) such that

∥(λ′ ∘ ϕ|_{A^{i}})(f) − u_{i}(λ′ ∘ ψ|_{A^{i}})(f)u_{i}^{∗}∥ < 6ε, ∀f ∈ F_{i}.

Then the unitary u = ⨁_{i} u_{i} ⊕ (1 − ∑_{i} λ′(E_{i})) satisfies

∥(λ′ ∘ ϕ)(f) − u(λ′ ∘ ψ)(f)u^{∗}∥ < 6ε, ∀f ∈ F.

Hence

∥(λ ∘ ϕ)(f) − u(λ ∘ ψ)(f)u^{∗}∥ < 6ε + 2ε = 8ε, ∀f ∈ F.
⊓ ⊔

Remark 1.6.10. The version of Theorem 2.29 of [EG2] with A being a direct sum of full matrix algebras is a direct consequence of the above theorem and Corollary 2.24 of [EG2] (see [EG2, Theorem 2.21] also). In order to obtain the general version of Theorem 2.29 of [EG2], one needs to apply the dilation lemma [EG2, 2.13] and Lemma 1.6.8 above. (The number 8ε should be changed to 5 · 8ε = 40ε which is still better than 70ε in [EG2].) The following lemma is a direct consequence of Lemma 1.6.5.

Lemma 1.6.11. Let X be a finite simplicial complex and A = C(X). Let F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number δ > 0 with the following property. If B is a unital C^{∗}-algebra, ϕ_{t} : A → B, 0 ≤ t ≤ 1 is a continuous path of G-δ multiplicative maps (i.e., ϕ_{t} ∈ Map_{G−δ}(A, B)), then there are a positive integer L, a homomorphism λ : A → M_{L}(B) with finite dimensional image, and a unitary u ∈ M_{L+1}(B) such that

∥(ϕ_{0} ⊕ λ)(f) − u(ϕ_{1} ⊕ λ)(f)u^{∗}∥ < ε, ∀f ∈ F.

The proof of the following corollary has some similarity to the proof of Corollary 1.6.6. Such method will be used frequently.

Corollary 1.6.12. Let X be a finite simplicial complex and A = C(X). Let F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number δ > 0 with the following property. If B is a unital C^{∗}-algebra, p ∈ B is a projection, ϕ_{t} : A → pBp, 0 ≤ t ≤ 1 is a continuous path of G-δ multiplicative maps , then there are a positive integer L and a number η > 0 such that for any η-dense subset {x_{1}, x_{2}, ⋯, x_{•}} ⊂ X, any set of mutually orthogonal projections {p_{1}, p_{2}, ⋯, p_{•}} ⊂ B⊗K with [p_{i}] ≥ L·[p] and p_{i}⊥p, we have

∥ϕ_{0}(f) ⊕ ∑_{i=1}^{•} f(x_{i})p_{i} − u(ϕ_{1}(f) ⊕ ∑_{i=1}^{•} f(x_{i})p_{i})u^{∗}∥ < ε, ∀f ∈ F,

for a certain unitary u ∈ (p ⊕ p_{1} ⊕ ⋯ p_{•})(B ⊗ K)(p ⊕ p_{1} ⊕ ⋯ p_{•}).

Proof: For the finite set F ⊂ A, choose η small enough such that if dist(x, x′) < η, then ∥f(x) − f(x′)∥ < \frac{ε}{3} for all f ∈ F. Apply Lemma 1.6.11 to F and \frac{ε}{3} to obtain G and δ. For the path ϕ_{t} : A → pBp, there exist a positive integer L, a homomorphism λ : A → M_{L}(pBp), and a unitary u_{1} ∈ M_{L+1}(pBp) as in Lemma 1.6.11. That is

∥(ϕ_{0} ⊕ λ)(f) − u_{1}(ϕ_{1} ⊕ λ)(f)u_{1}^{∗}∥ < \frac{ε}{3}, ∀f ∈ F.

From 1.6.4, λ is of the form

λ(f) = ∑_{i=1}^{l} f(y_{i})q_{i},

where {y_{1}, y_{2}, ⋯, y_{l}} ⊂ X, and {q_{1}, q_{2}, ⋯, q_{l}} ⊂ M_{L}(pBp) is a set of mutually orthogonal projections. Since {x_{1}, x_{2}, ⋯, x_{•}} is an η-dense subset of X, we can divide the set {y_{1}, y_{2}, ⋯, y_{l}} into a disjoint union of subsets X_{1} ∪X_{2} ∪⋯∪X_{•} (some X_{i} may be empty) such that dist(y, x_{i}) < η for any y ∈ X_{i}. Set p′_{i} := ∑_{y_{j}∈X_{i}} q_{j} and define λ′ : C(X) → M_{L}(pBp) by λ′(f) = ∑_{i=1}^{•} f(x_{i})p′_{i}. (Note that for some i, p′_{i} might be 0.) Then from the way η is chosen, we have

∥λ′(f) − λ(f)∥ < \frac{ε}{3}, ∀f ∈ F.

Therefore,

∥(ϕ_{0} ⊕ λ′)(f) − u_{1}(ϕ_{1} ⊕ λ′)(f)u_{1}^{∗}∥ < ε, ∀f ∈ F.

Our corollary follows from the fact [p′_{i}] ≤ L · [p] ≤ p_{i}.

⊓ ⊔

If X does not contain any isolated point, then in the above corollary, we can change the condition [p_{i}] ≥ L·[p] to a weaker condition [p_{i}] ≥ [p], by choosing η smaller. (Roughly speaking, this is true because η could be chosen so small that if {x_{i}} is η-dense, then there are at least L points of x_{i} in the η′-neighborhood of any point in X for a pre-given small number η′. If X is a space of single point, then this is not true.) Therefore, the number L does not appear in the following corollary.

Corollary 1.6.13. Let X be a finite simplicial complex without any single point components, A = C(X). Let F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number δ > 0 with the following property. If B is a unital C^{∗}-algebra, p ∈ B is a projection, ϕ_{t} : A → pBp, 0 ≤ t ≤ 1, is a continuous path of G-δ multiplicative maps , then there is a number η > 0 such that for any η-dense subset {x_{1}, x_{2}, ⋯, x_{•}} ⊂ X, any set of mutually orthogonal projections {p_{1}, p_{2}, ⋯, p_{•}} ⊂ B ⊗ K with [p_{i}] ≥ [p] and p_{i}⊥p, we have

∥ϕ_{0}(f) ⊕ ∑_{i=1}^{•} f(x_{i})p_{i} − u(ϕ_{1}(f) ⊕ ∑_{i=1}^{•} f(x_{i})p_{i})u^{∗}∥ < ε, ∀f ∈ F

for a certain unitary u ∈ (p ⊕ p_{1} ⊕ ⋯ p_{•})(B ⊗ K)(p ⊕ p_{1} ⊕ ⋯ p_{•}).

Proof: Let L and η_{1} (in place of η) be as in Corollary 1.6.12. Let η_{2} be the minimum of the diameters of path connected components of X, which is positive since X has no single point component. And let η_{3} be a positive number such that if dist(x, x′) < η_{3}, then ∥f(x) − f(x′)∥ < ε. Define η′ = min(η_{1}, η_{2}, η_{3}). Let η = \frac{η′}{8L}. Suppose that X′ = {x_{1}, x_{2}, ⋯, x_{•}} is an η-dense finite subset of X. Choose a η′-dense subset \{x_{k_{1}}, x_{k_{2}}, ⋯, x_{k_{l}}\} ⊂ X′ such that dist(x_{k_{i}}, x_{k_{j}}) ≥ \frac{η′}{2} if i ≠ j. (Such subset exists. It could be chosen to be a maximum subset of X′ such that the distance of any two points in the set is at least \frac{η′}{2} . Then the η′-density follows from the maximality.) It is easy to see that there is a partition of X′ as X′ = X_{1} ∪ X_{2} ∪ ⋯ ∪ X_{l} such that

X′ ∩ B_{\frac{η′}{4}} (x_{k_{i}}) ⊂ X_{i} ⊂ X′ ∩ B_{η′}(x_{k_{i}}).

Since X′ is η-dense and η = \frac{η′}{8L},

#(X_{i}) ≥ #(X′ ∩ B_{\frac{η′}{4}} (x_{k_{i}})) ≥ L.

(Here we also use the fact that the connected component of x_{k_{i}} in X has diameter at least η′. ) Let p_{j}, j = 1, 2, ⋯ •, be the projections as in the corollary. Define q_{i} = ∑_{x_{j}∈X_{i}} p_{j}, i = 1, 2, ⋯ l. Then from [p_{j}] ≥ [p] and #(X_{i}) ≥ L, we have, [q_{i}] ≥ L · [p]. Our corollary (with 3ε in place of ε) follows from an application of Corollary 1.6.12 to {x_{k_{i}}}_{i=1}^{l} and {q_{i}}_{i=1}^{l}, and the estimation

∥ ∑_{i=1}^{•} f(x_{i})p_{i} − ∑_{i=1}^{l} f(x_{k_{i}})q_{i}∥ < ε, ∀f ∈ F.

(The above estimation is a consequence of the way η_{3} is chosen and the fact that X_{i} ⊂ B_{η′}(x_{k_{i}}) with η′ < η_{3}.)

⊓ ⊔

The following lemma is proved by applying Lemma 1.6.8.

Lemma 1.6.14. Let A = ⨁ A^{k} = ⨁_{k=1}^{l} M_{s(k)}(C(X_{k})), where X_{k} are connected simplicial complexes and {s(k)} are positive integers. For any finite set G′ ⊂ A, any number δ′ > 0, any finite sets G_{1}^{k} ⊂ C(X_{k}) and any numbers δ_{1}^{k} > 0, k = 1, 2, ⋯ l, there are a finite set G ⊂ A and a number δ > 0 such that if ϕ ∈ Map(A, B) is G-δ multiplicative, then there is a map ϕ′ ∈ Map(A, B) satisfying the following conditions.

(1) ϕ′ is G′-δ′ multiplicative;

(2) ∥ϕ′(g) − ϕ(g)∥ < δ′ for all g ∈ G;

(3) {ϕ′(1_{A^{k}})}_{k=1}^{l} are mutually orthogonal projections in B and ϕ′(e_{11}^{k}) ∈ B are subprojections of ϕ′(1_{A^{k}}) ∈ B. And if each ϕ_{1}^{k} ∈ Map(C(X_{k}), B) is the restriction of ϕ′ on e_{11}^{k}M_{s(k)}(C(X_{k}))e_{11}^{k} ≅ C(X_{k}), then one can identify ϕ′(1_{A^{k}})Bϕ′(1_{A^{k}}) with ϕ′(e_{11}^{k})Bϕ′(e_{11}^{k}) ⊗ M_{s(k)} such that

ϕ′ = ⨁_{k=1}^{l} ϕ_{1}^{k} ⊗ id_{s(k)}.

Furthermore, ϕ_{1}^{k} is G_{1}^{k}-δ_{1}^{k} Multiplicative.

Proof: The part of G_{1}^{k}-δ_{1}^{k} multiplicativity of ϕ_{1}^{k} follows from the G′-δ′ multiplicativity of ϕ′ if we enlarge the set G′ and reduce the number δ′ so that G′ ⊃ {g · e_{11}^{k} | g ∈ G_{1}^{k}} and δ′ < δ_{1}^{k}. Also we can assume that G′ contains {e_{ij}^{k}}—the set of all matrix units. By Lemma 1.6.8, without loss of generality, we assume that the restriction ϕ|_{⨁_{k=1}^{l} M_{s(k)}(C^{|} )} is a homomorphism.

Let ϕ_{1}^{k} = ϕ|_{e_{11}^{k}Ae_{11}^{k}} ∈ Map(C(X_{k}), ϕ(e_{11}^{k})Bϕ(e_{11}^{k})). Then ϕ′ := ⨁_{k=1}^{l} ϕ_{1}^{k}⊗id_{s(k)} is defined by

ϕ′(f) = ∑_{i,j} ϕ(e_{i1}^{k})ϕ(f_{ij} · e_{11}^{k})ϕ(e_{1j}^{k}),

where f = (f_{ij})_{s(k)×s(k)} ∈ A^{k}. Note that for the above f ∈ A^{k}, one can write f = ∑_{i,j} e_{i1}^{k} · (f_{ij} · e_{11}^{k}) · e_{1j}^{k}. Obviously, if we choose G to be the set of all the elements which can be expressed as products of at most ten elements from the set G′, and if we choose δ small enough, then (1) and (2) will hold for ϕ′. (Notice that G′ contains all the matrix units.)

⊓ ⊔

The following result follows from Lemma 1.6.14 and Corollary 1.6.12 (see also 1.2.19).

Corollary 1.6.15. Let A = ⨁_{k=1}^{l} M_{s(k)}(C(X_{k})), where X_{k} are connected finite simplicial complexes and s(k) are positive integers. Let F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number δ > 0 with the following property.

If B is a unital C^{∗}-algebra, p ∈ B is a projection, ϕ_{t} : A → pBp, 0 ≤ t ≤ 1 is a continuous path of G-δ multiplicative maps , then there are a positive integer L, and η > 0 such that for a homomorphism λ : A → B ⊗ K (with finite dimensional image), there is a unitary u ∈ B satisfying:

∥ϕ_{0}(f) ⊕ λ(f) − u(ϕ_{1}(f) ⊕ λ(f))u^{∗}∥ < ε, ∀f ∈ F,

provided that λ is of the following form: there are an η-dense subset {x_{1}, x_{2}, ⋯, x_{•}} ⊂ ∐_{k=1}^{l} X_{k} (= SP(A)), and a set of mutually orthogonal projections {p_{1}, p_{2}, ⋯, p_{•}} ⊂ λ(⨁_{k} e_{11}^{k})(B ⊗ K)λ(⨁_{k} e_{11}^{k}) with [p_{i}] ≥ L · [p], such that

λ(f) = ∑_{i=1}^{•} p_{i} ⊗ f(x_{i}), ∀f ∈ A

under the identification λ(1_{A^{k}})Bλ(1_{A^{k}}) ≅ (λ(e_{11}^{k})Bλ(e_{11}^{k})) ⊗ M_{s(k)}(C^{|} ).

Proof: One can apply Lemma 1.6.14 to ϕ_{t} ∈ Map(A, pBp[0, 1]) to reduce the problem to the case of A = C(X_{k}) which is Corollary 1.6.12. (Here pBp[0, 1] is defined to be the C^{∗}-algebra of continuous pBp valued functions on [0, 1].)

⊓ ⊔

For convenience, we introduce the following definitions.

Definition 1.6.16. A homomorphism ϕ : A = ⨁_{i=1}^{n} M_{k_{i}}(C(X_{i})) → B = ⨁_{j=1}^{n′} M_{l_{j}}(C(Y_{j})) is called m-large if for each partial map ϕ^{ij} : A^{i} = M_{k_{i}}(C(X_{i})) → B^{j} = M_{l_{j}}(C(Y_{j})) of ϕ,

rank(ϕ^{ij}(1_{A^{i}}) ≥ m · rank(1_{A^{i}})(= m · k_{i}).

Definition 1.6.17. Let X be a connected finite simplicial complex, A = M_{k}(C(X)). A unital *-monomorphism ϕ : A → M_{l}(A) is called a (unital) simple embedding if it is homotopic to the homomorphism id ⊕ λ, where λ : A → M_{l−1}(A) is defined by

λ(f) = diag(\underbrace{f(x_{0}), f(x_{0}), ⋯, f(x_{0})}_{l−1}),

for a fixed base point x_{0} ∈ X. Let A = ⨁_{i=1}^{n} M_{k_{i}}(C(X_{i})), where X_{i} are connected finite simplicial complexes. A unital *-monomorphism ϕ : A → M_{l}(A) is called a (unital) simple embedding, if ϕ is of the form ϕ = ⊕ϕ^{i} defined by

ϕ(f_{1}, f_{2}, ⋯, f_{n}) = (ϕ^{1}(f_{1}), ϕ^{2}(f_{2}), ⋯, ϕ^{n}(f_{n})),

where the homomorphisms ϕ^{i} : A^{i}(= M_{k_{i}}(C(X_{i}))) → M_{l}(A^{i}) are unital simple embeddings.

1.6.18. For each connected finite simplicial complex X, there is a three dimensional connected simplicial complex Y = Y_{1} ∨ Y_{2} ∨ ⋯ ∨ Y_{•} such that K^{∗}(X) = K^{∗}(Y), where Y_{i} are the following special spaces: [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞} and S^{2}. The space [0, 1] could be avoided in the construction of Y. But we would like to use the space [0, 1] for the following special case: If K^{0}(X) = ZZ and K^{1}(X) = 0 (e.g., X is contractible) and X is not the space of a single point, then we choose Y = [0, 1]. When X is the space of a single point, choose Y = {pt}. The following result is Lemma 2.1 of [D2] (see Lemma 3.13 and Lemma 3.14 of [G4] also).

Lemma 1.6.19. ([D2, 2.1]) Let B_{1} = ⨁_{j=1}^{s} M_{k(j)}(C(Y_{j})), where Y_{j} are the following spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}. Let X be a connected finite simplicial complex, let Y be the three dimensional space defined in 1.6.18 with K^{∗}(X) = K^{∗}(Y), and let A = M_{N}(C(X)). Let α_{1} : B_{1} → A be a homomorphism. For any finite sets G ⊂ B_{1} and F ⊂ A, and any number δ > 0, there exists a diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} , \end{matrix}

where A′ = M_{L}(A), B_{2} = M_{S}(C(Y)); ψ is a homomorphism, α_{2} is a unital homomorphism, and ϕ is a unital simple embedding (see 1.6.17); β ∈ Map(A, B_{2}) is F -δ multiplicative. Moreover there exist homotopies Ψ ∈ Map(B_{1}, B_{2}[0, 1]) and Φ ∈ Map(A, A′[0, 1]) such that Ψ is G-δ multiplicative, Φ is F -δ multiplicative, and

Ψ|_{1} = ψ, Ψ|_{0} = β ∘ α_{1}, Φ|_{0} = α_{2} ∘ β and Φ|_{1} = ϕ.

(In the application of this lemma, it is important to require that ϕ is a unital simple embedding. This requirement means that ϕ defines the same element in kk(X, X) (connective KK theory) as the identity map id : A → A. Roughly speaking, this lemma (and Theorem 6.26 below) means that an “identity map” could factor through matrix algebras over Y — a special space of dimension three.)

Proof: If one assumes that α_{1} : B_{1} → A is m-large (see 1.6.16) for a number m > 4 dim(X), then this lemma becomes Lemma 2.1 of [D2]. We make use of this special case to prove the general case as below.

Define a unital simple embedding λ : A → M_{m}(A) (m > 4 dim(Y)) by

λ(f) = diag(f, \underbrace{f(x_{0}), f(x_{0}), ⋯, f(x_{0})}_{m−1}).

Then α′_{1} = λ ∘ α_{1} is m-large. Apply Lemma 2.1 of [D2]— the special case of the lemma to α′_{1}, λ(F) ⊂ M_{m}(A) and G ⊂ B_{1} to obtain the following diagram

\begin{matrix}M_{m}(A) & ⟶^{ϕ′} & A′ \\ ↑_{α′_{1}} & ↘^{β′} & ↑_{α′_{2}} \\ B_{1} & ⟶^{ψ′} & B_{2} \end{matrix}

with homotopy paths Ψ′ and Φ′ with properties described in the lemma for the homomorphism α′_{1}, finite sets λ(F) ⊂ M_{m}(A) and G ⊂ B_{1}. Define β = β′ ∘ λ, ϕ = ϕ′ ∘ λ, α_{2} = α′_{2}, ψ = ψ′, Ψ = Ψ′ and Φ = Φ′ ∘ λ. Then we have the desired diagram with the desired properties.

⊓ ⊔

Remark 1.6.20. From the construction of ϕ and α_{2} in the proof of [D2, Lemma 2.1], we know that ϕ and α_{2} take trivial projections to trivial projections. But ψ may not take trivial projections to trivial projections unless α_{1} does.

1.6.21. Let X and Y be path connected finite simplicial complexes, and C = M_{k}(C(Y)), D = M_{l}(C(X)). Let x_{0} ∈ X and y_{0} ∈ Y be fixed base points. Then from Lemma 3.14 of [EG2], we have the following: any homomorphism ϕ ∈ Hom(C, D) is homotopy equivalent to a homomorphism ϕ′ ∈ Hom(C, D) (within Hom(C, D)) such that ϕ′(C^{0}) ⊂ D^{0}, where C^{0}, D^{0}, are the ideals of C and D, respectively, which consist of matrix valued functions vanishing on the base points (see 1.1.7(h)). In other words, there is a unitary U ∈ M_{l}(C^{|} ) such that

ϕ′(f)(x_{0}) = U ( \begin{matrix}f(y_{0}) \\ & . & . & . \\ & & & & f(y_{0}) \\ & & & & & 0 \\ & & & & & & . & . & . \\ & & & & & & & & & 0 \end{matrix} ) U^{∗} ∈ M_{l}(C^{|} ) ∀f ∈ C.

Notice that if a homomorphism α_{2} is as desired in Lemma 1.6.19, then any homomorphism, which is homotopic to α_{2}, is also as desired. Therefore, in Lemma 1.6.19, we can require that the homomorphism α_{2} : B_{2}(= M_{S}(C(Y)) → A′(= M_{L}(C(X))) is of the above form for certain base points y_{0} ∈ Y and x_{0} ∈ X.

In the following, let us explain why we can also choose the homomorphism α_{2} to be injective. If X is the space of a single point, then Y is also the space of a single point by our choice. And therefore, α_{2} is injective, since B_{2} is simple. If the connected simplicial complex X is not a single point (i.e., dim(X) ≥ 1), then it can be proved that there is a continuous surjective map g : X → Y, using the standard idea of Peano curve. In fact, one can assume that the map g is homotopy trivial—one can make it factor through an interval. On the other hand, by Theorem 6.4.4 of [DN], if L ≥ 3S(dim(X) + 1), then for the unital homomorphism α_{2} : M_{S}(C(Y)) → M_{L}(C(X)), there is a homomorphism α′ : M_{S}(C(Y)) → M_{L−S}(C(X)) such that α_{2} is homotopic to the homomorphism defined by

f ↦ diag(α′(f), f(y_{0})),

Then α_{2} is homotopic to diag(α′, g^{∗}) defined by

f ↦ diag(α′(f), f ∘ g),

since g is homotopy trivial. So we can replace α_{2} by diag(α′, g^{∗}) which is injective, since g is surjective. Similarly, if SP(B_{2}) = Y is not a single point space (i.e., X is not the space of a single point), then the homomorphism ψ : B_{1} → B_{2} could be chosen to be injective with in the same homotopy class of Hom(B_{1}, B_{2}), provided that α_{1}(1_{B_{1}^{i}}) ≠ 0, for each block B_{1}^{i} of B_{1} (later on, we will always assume α_{1} satisfies this condition, since otherwise this block can be deleted from B_{1}).

Lemma 1.6.22. Let B = M_{k}(C(Y)), A = M_{l}(C(X)). Suppose that a unital homomorphism α : B → A satisfies α(B^{0}) ⊂ A^{0}, and takes any trivial projections of B to trivial projections of A, where B^{0} = M_{k}(C_{0}(Y)) := {f ∈ M_{k}(C(Y)) | f(y_{0}) = 0}, and A^{0} = M_{l}(C_{0}(X)) := {f ∈ M_{l}(C(X)) | f(x_{0}) = 0}, for some fixed base points y_{0} ∈ Y and x_{0} ∈ X. Let β_{0} : B → M_{n}(B) and β_{1} : A → M_{n}(A) be unital homomorphisms defined by

β_{0}(f)(y) = diag(f(y), f(y_{0}), ⋯, f(y_{0})), ∀f ∈ B,

and

β_{1}(f)(x) = diag(f(x), f(x_{0}), ⋯, f(x_{0})), ∀f ∈ A.

Then the following diagram commutes up to unitary equivalence.

A ⟶^{β_{1}} M_{n}(A)
α↑  ↑_{α⊗idn}
B ⟶^{β_{0}} M_{n}(B).

I.e., there is a unitary u ∈ M_{n}(A) such that

β_{1} ∘ α = Adu ∘ (α ⊗ id_{n}) ∘ β_{0}.

Proof: β_{1} ∘ α is defined by:

f ↦ α(f) ↦ diag(α(f), α(f)(x_{0}), ⋯, α(f)(x_{0})),

and (α ⊗ id_{n}) ∘ β_{0} is defined by:

f ↦ diag(f, f(y_{0}), ⋯, f(y_{0})) ↦ diag(α(f), α(f(y_{0})), ⋯, α(f(y_{0}))),

where α(f(y_{0})) denotes the result of α acting on the constant function g = f(y_{0}). On the other hand, from α(B^{0}) ⊂ A^{0}, we get

α(f)(x_{0}) ∼_{u} diag(\underbrace{f(y_{0}), ⋯, f(y_{0})}_{\frac{l}{k}}),

and from the fact that α takes trivial projections to trivial projections, we get

diag(\underbrace{f(y_{0}), ⋯, f(y_{0})}_{\frac{l}{k}}) ∼_{u} α(f(y_{0})),

where the symbol ∼_{u} means to be unitarily equivalent.

⊓ ⊔

The following result is from [EG2] (see 5.10, 5.11 of [EG2]).

Lemma 1.6.23. Let Y = Y_{1}∨Y_{2}∨⋯∨Y_{m}. If n is large enough, then any unital homomorphism β : M_{k}(C(Y)) → M_{nk}(C(Y)) is homotopic to a homomorphism β′ : M_{k}(C(Y)) → M_{nk}(C(Y)), which factors through ⨁_{i=1}^{m} M_{k_{i}}(C(Y_{i})), for certain integers {k_{i}}, as

β′ : M_{k}(C(Y)) ⟶^{β_{1}} ⨁_{i=1}^{m} M_{k_{i}}(C(Y_{i})) ⟶^{β_{2}} M_{nk}(C(Y)).

Furthermore, β_{1} and β_{2} above can be chosen to be injective.

Proof: If k = 1, then the lemma is Lemma 5.11 of [EG2]. (Notice that, we choose both spaces X and Y in Lemma 5.11 of [EG2] to be the above space Y. In addition, the spaces X_{i} in Lemma 5.11 of [EG2] could be chosen to be spaces Y_{i} in our case, according to 5.10 of [EG2].) For the general case, one writes β as b ⊗ id_{k}, where b = β|_{e_{11}M_{k}(C(Y))e_{11}} : C(Y) → β(e_{11})M_{nk}(C(Y))β(e_{11}), then apply Lemma 5.11 of [EG2] to b. Furthermore, one can make β_{1} and β_{2} injective in the same way as in the end of 1.6.21. (Or one observes that the maps β_{1} and β_{2} constructed in Lemma 5.11 of [EG2] are already injective for our case.)

⊓ ⊔

Combining Lemmas 1.6.19, 1.6.21, 1.6.22 and 1.6.23, we have the following Lemma:

Lemma 1.6.24. Let B_{1} = ⨁_{j=1}^{s} M_{k(j)}(C(Y_{j})), where Y_{j} are spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}. Let X be a connected finite simplicial complex and let A = M_{N}(C(X)). Let α_{1} : B_{1} → A be a homomorphism with α_{1}(1_{B_{1}^{i}}) ≠ 0 for each block B_{1}^{i} of B_{1}. For any finite sets G ⊂ B_{1} and F ⊂ A, and any number δ > 0, there exists a diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} , \end{matrix}

where A′ = M_{L}(A), and B_{2} is a direct sum of matrix algebras over the spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}; ψ is a homomorphism, α_{2} is a unital injective homomorphism, and ϕ is a unital simple embedding (see 1.6.17). β ∈ Map(A, B_{2}) is F -δ multiplicative. Moreover, there exist homotopies Ψ ∈ Map(B_{1}, B_{2}[0, 1]) and Φ ∈ Map(A, A′[0, 1]) such that Ψ is G-δ multiplicative, Φ is F -δ multiplicative, and

Ψ|_{1} = ψ, Ψ|_{0} = β ∘ α_{1}, Φ|_{0} = α_{2} ∘ β and Φ|_{1} = ϕ.

Furthermore, if X is not the space of a single point, then at least one of the blocks of B_{2} has spectrum different from the space of single point and ψ can be chosen to be injective.

Proof: Let

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} , \end{matrix}

be the diagram described in Lemma 1.6.19 with homotopies Φ and Ψ. Let n be the integer obtained by applying Lemma 1.6.23 to B_{2} = M_{k}(C(Y)). Then apply 1.6.22 to α_{2} : B_{2} → A′ to obtain a diagram

A′  ⟶^{β_{1}} M_{n}(A′)
α2↑  ↑_{α2⊗idn}
B_{2}  ⟶^{β_{0}} M_{n}(B_{2}).

which commutes up to homotopy. (Here we have the condition that α_{2} takes trivial projections to trivial projections from Remark 1.6.20. Also, α_{2} is homotopic to a homomorphism which takes B^{0} to A^{′0}.)

Furthermore, from Lemma 1.6.23, β_{0} is homotopic to a homomorphism β′_{0} factoring through a C^{∗}-algebra newB_{2} which is a direct sum of matrix algebras over spaces {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}. Now it is routine to finish the construction of the diagram. We omit the details.

⊓ ⊔

1.6.25. Our next task is to add a homomorphism λ : A → M_{n}(B_{2}) into the diagram in Lemma 1.6.24 to obtain diagrams,

A
↑_{α_{1}} ↘^{β⊕λ}
B_{1}  ⟶^{ψ⊕(λ∘α_{1})} M_{n+1}(B_{2})

and

A ⟶^{ϕ⊕((α_{2}⊗idn)∘λ)} M_{n+1}(A′)
↘^{β⊕λ} ↑_{α_{2}⊗id_{n+1}}
M_{n+1}(B_{2})

which are almost commutative up to unitary equivalence, using Corollary 1.6.15. To do so, we make the following assumption. Assumption: α_{1} : B_{1} → A is injective. Let G_{1} ⊂ B_{1} and F_{1} ⊂ A be any finite sets, and ε > 0. We will make the above diagrams approximately commute on G_{1} and F_{1}, respectively, to within ε, up to unitary equivalence. Apply Corollary 1.6.15 to G_{1} ⊂ B_{1} (in place of F ⊂ A) and ε > 0, to obtain G ⊂ B_{1} and δ_{1} (in place of the set G and the number δ, respectively in Corollary 1.6.15). Similarly, apply Corollary 1.6.15 to F_{1} ⊂ A (in place of F ⊂ A) and ε > 0 to obtain F ⊂ A and δ_{2} (in place of the set G and the number δ in Corollary 1.6.15). Let δ = min(δ_{1}, δ_{2}). Suppose that the diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} \end{matrix}

is the one constructed in Lemma 1.6.24 with homotopy path Ψ ∈ Map(B_{1}, B_{2}[0, 1]) between β ∘ α_{1} and ψ, and homotopy path Φ ∈ Map(A, A′[0, 1]) between α_{2} ∘ β and ϕ, corresponding to the sets G ⊂ B_{1}, F ⊂ A and the number δ > 0. Regarding the homotopy path Ψ as the homotopy path ϕ_{t} in Corollary 1.6.15, we can obtain η_{1}, L_{1} as the numbers η and L in Corollary 1.6,15. Similarly, replacing the above Ψ by Φ, we obtain η_{2}, L_{2}.

Notice that the injectivity of α_{1} implies that, for each block B_{1}^{j} of B_{1}, SP(α_{1}^{j}) = Y_{j}(= SP(B_{1}^{j})). Therefore there is an η_{2}-dense subset {x_{1}, x_{2}, ⋯ x_{m}} of X such that ⋃_{i=1}^{m} SPα_{1}^{j}|_{x_{i}} is η_{1}-dense in Y_{j} for each j ∈ {1, 2, ⋯, s}. Define λ_{1} : A(= M_{N}(C(X))) → M_{mN}(B_{2}) by

λ_{1}(f) = diag(1_{B_{2}} ⊗ f(x_{1}), 1_{B_{2}} ⊗ f(x_{2}), ⋯, 1_{B_{2}} ⊗ f(x_{m})).

Then λ_{1} ∘ α_{1} : B_{1} → M_{mN}(B_{2}) is a homomorphism defined by the point evaluations on the η_{1}-dense subset ⋃_{j=1}^{s} ⋃_{i=1}^{m} SPα_{1}^{j}|_{x_{i}} ⊂ SPB_{1}. Also (α_{2} ⊗ id_{mN}) ∘ λ_{1} : A → M_{mN}(A′) is defined by point evaluations on the η_{2}-dense subset {x_{j}}_{j=1}^{m} ⊂ X. Let L = max(L_{1}, L_{2}) and n = mNL. Define λ : A → M_{n}(B_{2}) =

M_{L}(M_{mN}(B_{2})) by λ = diag(\underbrace{λ_{1}, λ_{1}, ⋯, λ_{1}}_{L}).

Then, obviously, λ∘α_{1} : B_{1} → M_{n}(B_{2}) satisfies the condition for λ in Corollary 1.6.15, for the homotopy Ψ, positive integer L_{1}, and η_{1} > 0. And so does (α_{2} ⊗ id_{n}) ∘ λ : A → M_{n}(A′) for Φ, L_{2} and η_{2}. Therefore, there are unitaries u_{1} ∈ M_{n+1}(B_{2}) and u_{2} ∈ M_{n+1}(A′) such that

∥((β ⊕ λ) ∘ α_{1})(f) − u_{1}(ψ ⊕ (λ ∘ α_{1}))(f)u_{1}^{∗}∥ < ε, ∀f ∈ G_{1},

∥(ϕ ⊕ ((α_{2} ⊗ id_{n}) ∘ λ))(f) − u_{2}((α_{2} ⊗ id_{n+1}) ∘ (β ⊕ λ))(f)u_{2}^{∗}∥ < ε, ∀f ∈ F_{1}.

In the diagram in Lemma 1.6.24, if we replace B_{2} by M_{n+1}(B_{2}), A′ by M_{n+1}(A′), ψ by Adu_{1} ∘ (ψ ⊕ (λ ∘ α_{1})), β by β ⊕ λ, α_{2} by Adu_{2} ∘ (α_{2} ⊗ id_{n+1}), and finally ϕ by ϕ ⊕ ((α_{2} ⊗ id_{n}) ∘ λ), then we have the diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} \end{matrix}

for which, the lower left triangle is approximately commutative on G_{1} to within ε and the upper right triangle is approximately commutative on F_{1} to within ε. Since G_{1} and F_{1} are arbitrary finite subsets, we proved the following main factorization result.

Theorem 1.6.26. Let B_{1} = ⨁_{j=1}^{s} M_{k(j)}(C(Y_{j})), where Y_{j} are spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}. Let X be a connected finite simplicial complex and let A = M_{N}(C(X)). Let α_{1} : B_{1} → A be an injective homomorphism. For any finite sets G ⊂ B_{1} and F ⊂ A, and for any numbers ε > 0 and δ > 0 there exists a diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} , \end{matrix}

where A′ = M_{L}(A), and B_{2} is a direct sum of matrix algebras over the spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}; ψ is an injective homomorphism, α_{2} is a unital injective homomorphism, and ϕ is a unital simple embedding (see 1.6.17). β ∈ Map(A, B_{2}) is F -δ multiplicative. Moreover

∥ψ(f) − (β ∘ α_{1})(f)∥ < ε, ∀f ∈ G;
∥ϕ(f) − (α_{2} ∘ β)(f)∥ < ε, ∀f ∈ F.

Corollary 1.6.27. Theorem 1.6.26 still holds if one replaces the injectivity condition of α_{1} by the following condition: For each block B_{1}^{j} of B_{1}, either α_{1}^{j} is injective or α_{1}^{j}(B_{1}^{j}) is a finite dimensional subalgebra of A. (Of course, one still needs to assume that α_{1}(1_{B_{1}^{j}}) ≠ 0 for each block B_{1}^{j} of B_{1} and that at least one block of B_{2} has spectrum different from the space of single point (equivalently, X ≠ {pt}) , if he wants the homomorphism ψ to be injective. If one does not assume the above condition, he could still get the following dichotomy condition for ψ: For each block B_{1}^{j} of B_{1} and B_{2}^{k} of B_{2}, either ψ^{j,k} is injective or ψ^{j,k} has a finite dimensional image.)

Proof: Write B_{1} = B′ ⊕ B′′ such that α_{1} is injective on B′ and α_{1}(B′′) ⊂ A is of finite dimension. Consider the finite dimensional algebra

D := ⨁_{B_{1}^{i}⊂B′}(α_{1}(1_{B_{1}^{i}}) · C^{|} ) ⨁ α_{1}(B′′) ⊂ A.

By Lemma 1.6.8, if β : A → B_{2} is sufficiently multiplicative, then β is close to such a map β′ that the restriction β′|_{D} is a homomorphism. β′ can be connected to the original β by a linear path. If the original map β is sufficiently multiplicative, then the connecting path, regarded as a map from A to B_{2}[0, 1], is F -δ multiplicative for any pre-given finite set F ⊂ A and number δ > 0. Therefore, with out loss of generality, we assume that β|_{D} is a homomorphism for the original map β in 1.6.24. By Lemma 1.6.8 again, if Ψ : B_{1} → B_{2}[0, 1] is sufficiently multiplicative, then Ψ is close to a map Ψ′ such that Ψ′|_{r(B_{1})} is a homomorphism, where r(B_{1}) is defined in 1.1.7(h). Note that Ψ|_{1} = ψ is a homomorphism and (Ψ|_{0})|_{r(B_{1})} = β|_{D} ∘ (α_{1}|_{r(B_{1})}) is also a homomorphism. From the proof of Lemma 1.6.8, we can see that the above Ψ′ can be chosen such that Ψ′|_{1} = Ψ|_{1} and Ψ′|_{0} = Ψ|_{0}. Therefore, without loss of generality, we can assume that the homotopy path Ψ in Lemma 1.6.24 satisfies that Ψ|_{r(B_{1})} is a homomorphism.

Up to a unitary equivalence, we can further assume that Ψ_{t}(1_{B_{1}^{i}}) = Ψ_{t′}(1_{B_{1}^{i}}) for any t, t′ ∈ [0, 1] and any block B_{1}^{i} of B_{1}. One can repeat the procedure in 1.6.25 to construct the homomorphism λ : A → M_{n}(A), defined by point evaluations on an η_{2}-dense subset {x_{1}, x_{2}, ⋯, x_{m}} ⊂ X, to satisfy the condition that λ ∘ α_{1}^{j} is defined by point evaluations on an η_{1}-dense subset ⋃_{i=1}^{m} SPα_{1}^{j}|_{x_{i}} ⊂ SPB_{1}^{j} of sufficiently large size, for each block B_{1}^{j} of the part B′. As in 1.6.25, we can define newβ to be β ⊕ λ. At the same time, ϕ and α_{2} can also be defined as in 1.6.25. To define ψ, we need to consider two cases. For the blocks B_{1}^{j} in B′, ψ can be defined as in 1.6.25, since λ ∘ α_{1}^{j} is defined by point evaluations on an η_{1}-dense subset (of sufficiently large size). For the blocks B_{1}^{j} ⊂ B′′, we define ψ to be (β ⊕ λ) ∘ α_{1}^{j} = (newβ) ∘ α_{1}^{j}. (Note that β|_{α_{1}(B′′)} is a homomorphism.)

⊓ ⊔

Remark 1.6.28. Once the diagram in Theorem 1.6.26 (or Corollary 1.6.27) exists for A′ = M_{L}(A), then for any L′ > L, one can construct a diagram with the same property as in the theorem or the corollary with A′ = M_{L′}(A). This is easily seen from the following. Let r(A) = M_{N}(C^{|} ) and r : A → r(A) be as in 1.1.7(h). Let newB_{2} = oldB_{2} ⊕ r(A), newβ = oldβ ⊕ r, newϕ = diag(oldϕ, \underbrace{i ∘ r, ⋯, i ∘ r}_{L′−L}), newψ =

oldψ ⊕ (r ∘ α_{1}), and newα_{2} = oldα_{2} ⊕ diag(\underbrace{i, ⋯, i}_{L′−L}), where i : r(A) → A ⊂

M_{L′−L}(A) ⊂ M_{L′}(A) is the inclusion (note that r(A) is a subalgebra of A as in 1.1.7(h)), and oldB_{2}, oldβ, oldϕ and oldα_{2} are B_{2}, β, ϕ and α_{2}, respectively, from Lemma 1.6.26 or Corollary 1.6.27.

Corollary 1.6.29. Let B_{1} = ⨁_{j=1}^{s} M_{k(j)}(C(Y_{j})), where Y_{j} are spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}. Let A = ⨁_{j=1}^{t} M_{l(j)}(C(X_{j})), where X_{j} are connected finite simplicial complex. Let α_{1} : B_{1} → A be a homomorphism satisfying the following condition: For each pair of blocks B_{1}^{i} of B_{1} and A^{j} of A, either the partial map α_{1}^{i,j} is injective or α_{1}^{i,j}(B_{1}^{i}) is a finite dimensional subalgebra of A^{j}. For any finite sets G ⊂ B_{1} and F ⊂ A, and for any numbers ε > 0 and δ > 0, there exists a diagram

\begin{matrix}A & ⟶^{ϕ} & A′ \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} \end{matrix}

where A′ = M_{L}(A), and B_{2} is a direct sum of matrix algebras over the spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{k=2}^{∞}, {T_{III,k}}_{k=2}^{∞}, and S^{2}; ψ is a homomorphism, α_{2} is a unital injective homomorphism, and ϕ is a unital simple embedding (see 1.6.17). β ∈ Map(A, B_{2}) is F -δ multiplicative. Moreover,

∥ψ(f) − (β ∘ α_{1})(f)∥ < ε, ∀f ∈ G;
∥ϕ(f) − (α_{2} ∘ β)(f)∥ < ε, ∀f ∈ F.

If we further assume that α_{1} satisfies the condition that α_{1}^{i,j}(1_{B_{1}^{i}}) ≠ 0 ∈ A^{j} for any partial map α_{1}^{i,j} : B_{1}^{i} → A^{j} of α_{1}, then either the homomorphism ψ could be chosen to be injective, or the spectra of all blocks of B_{2} could be chosen to be the spaces of a single point.

Proof: We can construct the diagram for each block A^{j} of A, then put them together in the obvious way. Using Remark 1.6.28, we can assume for each block A^{j}, A′_{j} = M_{L}(A_{j}) for the same L.

⊓ ⊔

The following is Lemma 4.6 of [G4] (see Lemma 1.2 of [D2]).

Lemma 1.6.30. Let A = ⨁_{i=1}^{t} M_{l(i)}(C(X_{i})), where X_{i} are connected finite simplicial complexes. Let A′ = M_{L}(A). Let the algebra r(A) and the homomorphism r : A → r(A) be as in 1.1.7(h). Let B be a direct sum of matrix algebras over finite simplicial complexes of dimension at most m. Let ϕ : A → A′ be a unital simple embedding (see Definition 1.6.17). For any (not necessarily unital) (m · L)-large homomorphism ϕ′ : A → B, there is a homomorphism λ : A′ ⊕ r(A) → B such that ϕ′ is homotopic to λ ∘ (ϕ ⊕ r). Furthermore, λ could be chosen to satisfy the condition that for any block B^{j} with SP(B^{j}) ≠ {pt}, the partial map λ^{·,j} : A′ ⊕ r(A) → B^{j} of λ is injective as remarked in 1.6.21. (This lemma will be applied in conjunction with Lemma 1.6.26 or Corollary 1.6.29. From here, one can see the importance of the requirement that ϕ is a unital simple embedding.)

Remark 1.6.31. In order to apply Lemma 1.6.30 later, we would like to do one more modification for Corollary 1.6.29. Let r : A → r(A) be as in 1.1.7(h). Then the diagram in Corollary 1.6.29 could be modified to the following diagram

\begin{matrix}A & ⟶^{ϕ⊕r} & A′ ⊕ r(A) \\ ↑_{α_{1}} & ↘^{β⊕r} & ↑_{α_{2}⊕id} \\ B_{1} & ⟶^{ψ⊕(r∘α_{1})} & B_{2} ⊕ r(A) \end{matrix}

which satisfies that β ⊕ r is F -δ multiplicative and

∥(ψ ⊕ (r ∘ α_{1}))(f) − ((β ⊕ r) ∘ α_{1})(f)∥ < ε, ∀f ∈ G;
∥(ϕ ⊕ r)(f) − ((α_{2} ⊕ id) ∘ (β ⊕ r))(f)∥ < ε, ∀f ∈ F.

In the application of 1.6.29 and 1.6.30 in the proof of our main reduction theorem, we will still denote B_{2} ⊕ r(A) by B_{2}, β ⊕ r by β, ψ ⊕ (r ∘ α_{1}) by ψ, and α_{2} ⊕ id by α_{2}. So the diagram is

\begin{matrix}A & ⟶^{ϕ⊕r} & A′ ⊕ r(A) \\ ↑_{α_{1}} & ↘^{β} & ↑_{α_{2}} \\ B_{1} & ⟶^{ψ} & B_{2} . \end{matrix}

2 Spectral Multiplicity

In this section, we will show how to perturb a homomorphism ϕ : C(X) → P M_{k}(C(Y))P in such a way that the resulting homomorphism does not have large spectral multiplicities (see 1.2.4). Namely, the following result will be proved.

Theorem 2.1. Let X and Y be connected simplicial complexes with X ≠ {pt}. Set dim(X) + dim(Y) = l. For any given finite set G ⊂ C(X), any ε > 0, and any unital homomorphism ϕ : C(X) → P M_{•}(C(Y))P, where P ∈ M_{•}(C(Y)) is a projection, there is a unital homomorphism ϕ′ : C(X) → P M_{•}(C(Y))P such that

(1) ∥ϕ(g) − ϕ′(g)∥ < ε for all g ∈ G, and

(2) ϕ′ has maximum spectral multiplicity at most l.

2.2. Let k be a positive integer. Let Hom(C(X), M_{k}(C^{|} ))_{1} = F^{k}X. The space F^{k}X is compact and metrizable. We can endow the space F^{k}X with a fixed metric d as below. Choose a finite set {f_{i}}_{i=1}^{n} ⊂ C(X) which generates C(X) as a C^{∗}-algebra (e.g. one can embed X into IR^{n}, then choose {f_{i}} to be the coordinate functions). For any ϕ, ψ ∈ F^{k}X which, by definition, are unital homomorphisms from C(X) to M_{k}(C^{|} ), define

d(ϕ, ψ) = ∑_{i=1}^{n} ∥ϕ(f_{i}) − ψ(f_{i})∥.

Without loss of generality, we can assume that the above finite set {f_{i}}_{i=1}^{n} ⊂ G. On the other hand, F^{k}X is a finite simplicial complex (see [DN], [Se] and [Bl1]).

2.3. Let k = rank(P), where P is the projection in Theorem 2.1. For any fixed y, there is a unitary u_{y} ∈ M_{•}(C^{|}) such that P (y) = u_{y}diag(1_{k}, 0)u_{y}^{∗} (as in 1.2.1). Using this unitary, one can identify P (y)M_{•}(C^{|} )P (y) with M_{k}(C^{|} ) by sending a ∈ P (y)M_{•}(C^{|} )P (y) to the element in M_{k}(C^{|} ) corresponding to the upper left corner of u_{y}^{∗}au_{y}. (Notice that for any a ∈ P (y)M_{•}(C^{|} )P (y), the matrix

u_{y}^{∗}au_{y} = u_{y}^{∗}P (y)aP (y)u_{y} = u_{y}^{∗}P (y)u_{y}u_{y}^{∗}au_{y}u_{y}^{∗}P (y)u_{y}

= diag(1_{k}, 0)u_{y}^{∗}au_{y}diag(1_{k}, 0)

has the form

( \begin{matrix}(∗)_{k×k} & 0 \\ 0 & 0 \end{matrix} ) .)

In this way, for any y ∈ Y, the space Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1} can be identified with F^{k}X. Consider the disjoint union

⋃_{y∈Y} Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1}

as a subspace of Hom(C(X), M_{•}(C^{|} )) × Y with the induced topology. Using the above identification we can define a locally trivial fibre bundle

⋃_{y∈Y} Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1}

↓ π
Y

with fibre F^{k}X, as shown below, where π is the natural map sending any element in the set Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1} to the point y. For simplicity, write E_{P} := ⋃_{y∈Y} Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1}.

For any point y_{0} ∈ Y, there are an open set U ∋ y, and a continuous unitary valued function u : U → M_{•}(C^{|} ) such that P (y) = u(y)diag(1_{k}, 0)u^{∗}(y). (See 1.2.1.) Let R : M_{•}(C^{|} ) → M_{k}(C^{|} ) be the map taking any element in M_{•}(C^{|} ) to the k ×k upper left corner of the element. Let the trivialization t_{U} : π^{−1}(U) → U × F^{k}X be defined as follows. For any ϕ ∈ Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1} ⊂ π^{−1}(U), where y ∈ U, define t_{u}(ϕ) = (y, ψ), where ψ ∈ F^{k}X is defined by

ψ(f) = R(u^{∗}(y)ϕ(f)u(y)) for any f ∈ C(X).

(Again, u^{∗}(y)ϕ(f)u(y) is of the form ( \begin{matrix}(∗)_{k×k} & 0 \\ 0 & 0 \end{matrix} ) .)

Since the set Y is compact, there is a finite cover U = {U} of Y with the above trivialization for each U. This defines a fibre bundle π : E_{P} → Y. (See §1.4 for the definition and other materials of fibre bundle.)

2.4. In the above fibre bundle, the structure group Γ ⊂ Homeo(F^{k}X) could be chosen to be the collection of all γ ∈ Homeo(F^{k}X) of the form: there is a unitary u ∈ M_{k}(C^{|} ) such that

γ(ϕ)(f) = u^{∗}ϕ(f)u for any ϕ ∈ F^{k}X and f ∈ C(X).

One can see this as follows.

Suppose that U and V are two open sets in U, and t_{U} and t_{V} are trivializations, as in 2.3, defined by unitary valued functions u : U → M_{•}(C^{|} ) and v : V → M_{•}(C^{|} ), respectively. For any point y ∈ U ∩ V, the map t_{U} ∘ t_{V}^{−1} : F^{k}X → F^{k}X, can be computed as below. For any ϕ ∈ F^{k}X, define ϕ̃: C(X) → M_{•}(C^{|} ) by

ϕ̃(f) = ( \begin{matrix}ϕ(f)_{k×k} & 0 \\ 0 & 0 \end{matrix} ) , ∀f ∈ C(X).

Then

t_{V}^{−1}(ϕ)(f) = v(y)ϕ̃(f)v^{∗}(y) ∈ P (y)M_{•}(C^{|} )P (y),

and

t_{U} ∘ t_{V}^{−1}(ϕ)(f) = R(u^{∗}(y)v(y)ϕ̃(f)v^{∗}(y)u(y)).

Notice that

u(y)diag(1_{k}, 0)u^{∗}(y) = P (y) = v(y)diag(1_{k}, 0)v^{∗}(y).

It follows that v^{∗}(y)u(y) commutes with diag(1_{k}, 0). This implies that this matrix has the form

( \begin{matrix}(w_{1})_{k×k} & 0 \\ 0 & w_{2} \end{matrix} ) ,

where both w_{1} and w_{2} are unitaries. This shows that

t_{U} ∘ t_{V}^{−1}(ϕ)(f) = w_{1}^{∗}ϕ(f)w_{1}, ∀ϕ ∈ F^{k}X, f ∈ C(X).

In other words, t_{U} ∘ t_{V}^{−1} ∈ Γ. Obviously, Hom(C(X), P M_{•}(C(Y))P)_{1} can be regarded as a collection of continuous cross sections of the bundle π : E_{P} → Y. Since for any elements a, b ∈ M_{k}(C^{|} ) and unitary u ∈ M_{k}(C^{|} ),

∥uau^{∗} − ubu^{∗}∥ = ∥a − b∥,

it is easy to see that the metric d on F^{k}X defined in 2.2 is invariant under the action of any element in Γ in the sense of 1.4.6.

2.5. There is a natural map

θ : F^{k}X ⟶ P^{k}X

defined as follows. For any ϕ ∈ F^{k}X given by ϕ : C(X) → M_{k}(C^{|} ), define

θ(ϕ) = SP(ϕ) ∈ P^{k}X,

counting multiplicities. (See 1.2.5 and 1.2.7.) For each point x = [x_{1}, x_{2}, ⋯, x_{k}] ∈ P^{k}X, if the element x_{i} appears μ_{i} times in x for i = 1, 2, ⋯, k, then the maximum multiplicity of x is defined to be the maximum of μ_{1}, μ_{2}, ⋯, μ_{k}. The maximum multiplicity of a point ϕ ∈ F^{k}X is defined to be the maximum multiplicity of θ(ϕ) ∈ P^{k}X, which agrees with the maximum multiplicity of homomorphism ϕ : C(X) → M_{k}(C^{|} ) defined in 1.2.4. The homomorphism ϕ ∈ Hom(C(X), P M_{•}(C(Y))P)_{1} corresponds to a continuous cross section f : Y → E_{P}. This correspondence is one to one. For any cross section f : Y → E_{P}, any point y ∈ Y, the maximum multiplicity of f(y) is understood to be that obtained by regarding f(y) as an element in F^{k}X by an identification of Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1} with F^{k}X. Note that the maximum multiplicity of an element ϕ ∈ F^{k}X is invariant under the action of any element of Γ.

2.6. It is easy to see that for any finite set G ⊂ C(X) and ε > 0, there is an ε′ > 0 such that if d(ϕ_{y}, ϕ′_{y}) < ε′ for any y ∈ Y, then ∥ϕ(g) − ϕ′(g)∥ < ε for any g ∈ G, where ϕ_{y}, ϕ′_{y} ∈ F^{k}X are determined by an identification of Hom(C(X), P (y)M_{•}(C^{|} )P (y))_{1} with F^{k}X, as above. (Again the choice of the identification is not important, because the metric d is invariant under the action of any element in Γ.) Before proving Theorem 2.1, we prove the following weak version of Theorem 2.1.

Lemma 2.7. Let X and Y be as in Theorem 2.1. Let k > l = dim(X)+dim(Y). For any given finite set G ⊂ C(X), any ε > 0, and any unital homomorphism ϕ : C(X) → P M_{•}(C(Y))P, where P ∈ M_{•}(C(Y)) is a projection with rank(P) = k, there is a unital homomorphism ϕ′ : C(X) → P M_{•}(C(Y))P such that

(1) ∥ϕ(g) − ϕ′(g)∥ < ε for all g ∈ G, and

(2) ϕ′ has maximum spectral multiplicity at most k − 1. Comparing with Theorem 2.1, in the above result, we allow the maximum spectral multiplicity of the resulting homomorphism to be larger than l— only require it to be smaller than k = rank(P)—the maximum possible multiplicity. Since we assume that all the generators f_{i} of C(X) are inside the set G, Lemma 2.7 is equivalent to the following theorem.

Lemma 2.8. Suppose that X, Y, and P are as in Theorem 2.1 and that rank(P) = k > dim(X) + dim(Y). For any ε > 0 and any cross section f : Y → E_{P}, there is a cross section f′ : Y → E_{P} such that

(1) d(f(y), f′(y)) < ε for all y ∈ Y, and

(2) f′(y) has multiplicity at most k − 1 for all y ∈ Y. To prove our main theorem of this section—Theorem 2.1, we need the following result. The proof of this result will be given after the proof of Theorem 2.1.

Theorem 2.9. Suppose that X is a connected simplicial complex and X ≠ {pt}. For any ε > 0 and any x ∈ F^{m}X, there is a contractible open neighborhood U_{x} ∋ x with U_{x} ⊂ B_{ε}(x) ⊂ F^{m}X such that

π_{i}(U_{x}\{x}) = 0

for any 0 ≤ i ≤ m − 2. In other words, F^{m}X has property D(m) as in 1.4.3. We will use Theorem 2.9 and Corollary 1.4.10 (see also Theorem 1.4.9) to prove the following relative version of Lemma 2.8 (which gives rise to Lemma 2.8, by taking Y_{1} = ∅ ).

Lemma 2.10. Let X, Y, and P be as in Theorem 2.1, and Y_{1} ⊂ Y be a sub simplicial complex. Suppose that rank(P) = k > dim(X) + dim(Y). Suppose that a cross section f : Y → E_{P} satisfies the condition that f(y) has multiplicity at most k − 1, for any y ∈ Y_{1}. It follows that for any ε > 0, there is a cross section f′ : Y → E_{P} such that

(1) d(f(y), f′(y)) < ε for all y ∈ Y, and

(2) f′(y) has multiplicity at most k − 1 for all y ∈ Y.

(3) f′(y) = f(y), for any y ∈ Y_{1}.

Proof: Let F_{1} ⊂ F^{k}X denote the subset of all elements of maximum multiplicity equal to k. In other words, a homomorphism in F_{1} has one dimensional range. Obviously, F_{1} is the set of all homomorphisms ϕ ∈ Hom(C(X), M_{k}(C^{|} ))_{1} which are of the form

ϕ(f) = ( \begin{matrix}f(x) \\ & f(x) \\ & & . & . & . \\ & & & & & f(x) \end{matrix} )

for a certain point x ∈ X. Hence F_{1} is homeomorphic to X, and dim(F_{1}) = dim(X). As mentioned in 2.5, the maximum multiplicity of an element of F^{k}X is invariant under the action of Γ. So F_{1} is an invariant subset under the action of Γ. The conclusion of the Lemma 2.10 follows from Corollary 1.4.10 with E_{P} → Y, F^{k}X, F_{1}, Y_{1} and k in place of M → X, F, F_{1} , X_{1} and m in Corollary 1.4.10, respectively. (Note that, from Theorem 2.9, F^{k}X has property D(k).)

⊓ ⊔

The above lemma is equivalent to the following lemma (we stated it with projection Q instead of P to emphasis that we may use projections other than P —we will use subprojections of P).

Lemma 2.11. Let Y_{2} ⊂ Y_{1} ⊂ Y be sub-simplicial complexes of Y. Let Q ∈ M_{•}(C(Y_{1})) be a projection with rank m > l = dim(X)+dim(Y). For any given finite set G ⊂ C(X), any ε > 0, and any unital homomorphism ψ : C(X) → QM_{•}(C(Y_{1}))Q with the property that for any y ∈ Y_{2}, the multiplicity of ψ at y is at most m−1, there is a unital homomorphism ψ′ : C(X) → QM_{•}(C(Y_{1}))Q such that

(1) ∥ψ(g)(y) − ψ′(g)(y)∥ < ε for all g ∈ G and y ∈ Y_{1},

(2) ψ′ has spectral multiplicity at most m − 1,

(3) ψ′|_{Y_{2}} = ψ|_{Y_{2}}. In the proof of Theorem 2.1, we will not use Theorem 2.9 or Lemma 2.10 directly. We will use Lemma 2.11 instead. (So we do not need anything from fibre bundles in the rest of the proof of Theorem 2.1.)

2.12 Sketch of the idea of the proof of Theorem 2.1. Note that the proof of Lemma 2.10 can not be used to prove Theorem 2.1 (or the fibre bundle version of Theorem 2.1) in a straightforward way. For example, if we let F_{1} ⊂ F^{k}X be the subset of all elements with maximum multiplicity at least l + 1 (instead of k), then dim(F_{1}) may be very large— much larger than dim(X). In fact, dim(F_{1}) also depends on k and l. In Lemma 2.11 (or Lemma 2.10), we have already perturbed the homomorphism to avoid the largest possibility of maximum multiplicity—k. Next, we will perturb it again to avoid the next largest possibility of maximum multiplicity— k − 1. We will continue the procedure in this way. In general, suppose that the homomorphism ϕ : C(X) → P M_{•}(C(Y))P has maximum multiplicity m, with l < m ≤ k, we will prove that ϕ can be approximated arbitrarily well by another homomorphism ϕ′ with maximum multiplicity at most m − 1. Once this is done, Theorem 2.1 follows from a reverse induction argument beginning with m = k > l. (Note that for the case k ≤ l, we have nothing to prove.) To do the above, we need to work simplex by simplex. In fact, on each small simplex, the homomorphism ϕ can be decomposed into a direct sum of several homomorphisms ⨁_{i} ϕ_{i}, such that the projections ϕ_{i}(1) has rank at most m. Then we can apply Lemma 2.11 to each ϕ_{i} to avoid maximum multiplicity m. With these ideas in mind, it will not be difficult for the reader to construct the proof of Theorem 2.1. The complete detail will be contained in the next few lemmas, in particular, see the proof of Lemma 2.16.

Lemma 2.13. Suppose that P, X and Y are as in Theorem 2.1. For any ε > 0 and any positive integer d, there is a δ > 0 such that if Δ ⊂ Y is a simplex of dimension d, if P is regarded as a projection in M_{•}(C(Δ)), and if ψ : C(X) → P M_{•}(C(∂Δ))P is a homomorphism such that

∥ψ(g)(y) − ψ(g)(y′)∥ < δ, ∀g ∈ G, y, y′ ∈ ∂Δ,

then there is a homomorphism ψ′ : C(X) → P M_{•}(C(Δ))P such that

(1) ∥ψ′(g)(y) − ψ′(g)(y′)∥ < ε, ∀g ∈ G, y, y′ ∈ Δ, and

(2) ψ′|_{∂Δ} = ψ.

Proof: Note that P |_{Δ} is a trivial projection, since a simplex Δ is contractible.

So P M_{•}(C(Δ))P ≅ M_{k}(C(Δ)), where k = rank(P). The lemma follows from the fact that

F^{k}X = Hom(C(X), M_{k}(C^{|} ))_{1}

is a simplicial complex (see [DN] and [Bl]), which is locally contractible (see 1.4.2 and 1.4.3).

⊓ ⊔

We need the following lemma, which is obviously true.

Lemma 2.14. Suppose that ϕ : C(X) → P M_{•}(C(Y))P has maximum spectral multiplicity at most m. Then there exist η > 0 and δ > 0 such that the following statement holds. For any subset Z ⊂ Y with diameter(Z) < η, and homomorphism ψ : C(X) → P M_{•}(C(Z))P with the property that

∥ψ(g)(z) − ϕ(g)(z)∥ < δ, ∀z ∈ Z, g ∈ G,

there is a decomposition of ψ such as described below. There are open sets O_{1}, O_{2}, ⋯, O_{t} ⊂ X, with mutually disjoint closures (i.e., Ō_{i} ∩ Ō_{j} = ∅, ∀i ≠ j), and there are mutually orthogonal projections Q_{1}, Q_{2}, ⋯, Q_{t} ∈ M_{•}(C(Z)) and homomorphisms ψ_{i} : C(X) → Q_{i}M_{•}(C(Z))Q_{i} such that

1. ψ = ∑_{i=1}^{t} ψ_{i}

2. P (z) = ∑_{i=1}^{t} Q_{i}(z), ∀z ∈ Z,

3. rank(Q_{i}) ≤ m, and

4. SPψ_{i} ⊂ O_{i} for all i.

Proof: One can prove it using the following fact. Suppose that SPϕ ⊂ ∪O_{i}. If ψ is close enough to ϕ, then SPψ ⊂ ∪O_{i} and #(SPϕ ∩ O_{i}) = #(SPψ ∩ O_{i}) (see 1.2.12). Q_{i} in the lemma should be chosen to be the spectral projections of ψ corresponding to the open sets O_{i} (see 1.2.9). (Notice that Y is compact and that G ⊂ C(X) contains a set of the generators.)

⊓ ⊔

Lemma 2.15. Suppose that ϕ : C(X) → P M_{•}(C(Y))P has maximum spectral multiplicity at most m. Then there exist η > 0 and δ > 0 such that the following statement holds. For any ε > 0, any simplex Δ ⊂ Y (of any simplicial decomposition of Y) with diameter(Δ) < η, and any homomorphism ψ : C(X) → P M_{•}(C(Δ))P with the following properties:

(i) ∥ψ(g)(z) − ϕ(g)(z)∥ < δ, ∀z ∈ Δ, g ∈ G, and (ii) ψ|_{∂Δ} has maximum multiplicity at most m − 1, there exists a homomorphism ψ′ : C(X) → P M_{•}(C(Δ))P such that

(1) ∥ψ(g)(y) − ψ′(g)(y)∥ < ε for all g ∈ G and y ∈ Δ;

(2) ψ′ has spectral multiplicity at most m − 1; (3) ψ′|_{∂Δ} = ψ|_{∂Δ}.

Proof: . Suppose that η and δ are as in Lemma 2.14. If ψ is as described in this lemma, then one can obtain the decomposition ψ = ∑_{i=1}^{t} ψ_{i} of ψ as in Lemma 2.14. Then we only need to apply Lemma 2.11 to each map ψ_{i} to obtain ψ′_{i} : C(X) → Q_{i}M_{•}(C(Z))Q_{i} to satisfy the conclusion of Lemma 2.11 with ψ_{i}, Δ, ∂Δ, and Q_{i} in place of ψ, Y_{1}, Y_{2}, and Q, respectively. If ε is small enough, then SPψ′_{i} ⊂ O_{i}, where the open sets O_{i} are from Lemma 2.14. Hence the sum ψ′ = ∑_{i=1}^{t} ψ′_{i} is as desired.

⊓ ⊔

Lemma 2.16. Suppose that ϕ : C(X) → P M_{•}(C(Y))P has maximum spectral multiplicity at most m > l = dim(X) + dim(Y). For any simplicial subcomplex Y_{1} ⊂ Y, with respect to any simplicial decomposition of Y, and any ε > 0, there is a homomorphism ϕ′ : C(X) → P M_{•}(C(Y_{1}))P with multiplicity at most m−1 such that

∥ϕ′(g)(y) − ϕ(g)(y)∥ < ε, ∀g ∈ G, y ∈ Y_{1}.

(In particular, the above is true for Y_{1} = Y.)

Proof: We will prove the lemma by induction on dim(Y_{1}). If dim(Y_{1}) = 0, the lemma follows from the fact that, for a connected simplicial complex X with X ≠ {pt}, the subset of homomorphisms with distinct spectrum (maximum spectral multiplicity one) is dense in Hom(C(X), M_{k}(C^{|} ))_{1}. Suppose that the lemma is true for any simplicial subcomplex of dimension n, with respect to any simplicial decomposition. Let Y_{1} ⊂ Y be a simplicial complex of dimension n + 1 ≤ dim Y, with respect to some simplicial decomposition of Y. Let ε > 0. Let δ_{1}, η_{1} be the δ and η of Lemma 2.15. Apply Lemma 2.13 with n + 1 in place of d, and \frac{1}{4} min(ε, δ_{1}) in place of ε, to find δ_{2} as δ in the lemma. Choose η_{2} > 0 such that if dist(y, y′) < η_{2}, then

(∗)  ∥ϕ(g)(y) − ϕ(g)(y′)∥ < \frac{1}{4} min(ε, δ_{1}, δ_{2}), ∀g ∈ G.

Endow Y_{1} with a simplicial complex structure such that diameter(Δ) < min(η_{1}, η_{2}) for any simplex Δ of Y_{1}. Let Y′ ⊂ Y_{1} be the n-skeleton of Y_{1} with respect to the simplicial structure. From the inductive assumption, there is a homomorphism ϕ_{1} : C(X) → P M_{•}(C(Y′))P, with multiplicity at most m − 1, such that

(∗∗)  ∥ϕ_{1}(g)(y) − ϕ(g)(y)∥ < \frac{1}{4} min(ε, δ_{1}, δ_{2}), ∀g ∈ G and y ∈ Y′.(∗∗)

Consider a fixed simplex Δ ⊂ Y_{1} of top dimension (i.e., dim(Δ) = n + 1). Let us extend ϕ_{1}|_{∂Δ} to Δ (notice that ∂Δ ⊂ Y′). For any two points y, y′ ∈ ∂Δ, applying (*) to the pair of points y and y′, applying (**) to the points y and y′ separately, and combining all these three inequalities together, we get

∥ϕ_{1}(g)(y) − ϕ_{1}(g)(y′)∥ < \frac{3}{4} min(ε, δ_{1}, δ_{2}) < δ_{2}, ∀g ∈ G.

By Lemma 2.13 and the way δ_{2} is chosen, there is a homomorphism (let us still denote it by ϕ_{1}) ϕ_{1} : C(X) → P M_{•}(C(Δ))P, which extends the original ϕ_{1}|_{∂Δ}, such that

(∗∗∗) ∥ϕ_{1}(g)(y) − ϕ_{1}(g)(y′)∥ < \frac{1}{4} min(ε, δ_{1}), ∀g ∈ G and y, y′ ∈ Δ.

For any point y ∈ Δ, choose a point y′ ∈ ∂Δ. Applying both (*) and (***) to the pair (y, y′), applying (**) to the point y′ ∈ ∂Δ ⊂ Y′ and combining all these three inequalities together, we get

(∗∗∗∗)  ∥ϕ_{1}(g)(y) − ϕ(g)(y)∥ < \frac{3}{4} min(ε, δ_{1}), ∀g ∈ G and y ∈ Δ.

Since δ_{1} and η_{1} are chosen as in Lemma 2.15, and diameter(Δ) < η_{1}, it follows from (∗ ∗ ∗∗) and Lemma 2.15 that there is a homomorphism ϕ′ : C(X) → P M_{•}(C(Δ))P such that

(1) ∥ϕ′(g)(y) − ϕ_{1}(g)(y)∥ < \frac{1}{4}ε, ∀g ∈ G and y ∈ Δ.

(2) ϕ′ has spectral multiplicity at most m − 1.

(3) ϕ′|_{∂Δ} = ϕ_{1}|_{∂Δ}. Combining (1) above with (****), yields

∥ϕ′(g)(y) − ϕ(g)(y)∥ < \frac{3}{4} min(ε, δ_{1}) + \frac{1}{4}ε ≤ ε, ∀g ∈ G and y ∈ Δ.

Carry out the above construction independently for each simplex Δ. Since the definition of ϕ′ on ∂Δ is as same as ϕ_{1}, the definitions of ϕ′ on different simplices are agree on their intersection. By Lemma 1.2.14, this yields a homomorphism over the whole set Y_{1}. The lemma follows.

⊓ ⊔

Obviously, Theorem 2.1 follows from Lemma 2.16 by reverse induction argument beginning with m = k. (Note that we only need Lemma 2.16 for the case Y_{1} = Y.) Now we are going to prove Theorem 2.9, which is the only missing part in the proof of Theorem 2.1. The proof is somewhat similar to the proof of Theorem 6.4.2 of [DN]. It will therefore be convenient to recall some of the terminology and notation of [DN]. (It will be important to consider a certain method of decomposing the space F^{k}X.)

2.17. Recall from 6.17 of [DN] (cf. 1.2.4 above) that there is a map λ :

X^{k} × U(k) → F^{k}X, defined as follows. If u ∈ U(k) and (x_{1}, x_{2}, ⋯, x_{k}) ∈ X^{k}, then

(λ(x_{1}, x_{2}, ⋯, x_{k}, u))(f) = u ( \begin{matrix}f(x_{1}) \\ & f(x_{2}) \\ & & . & . & . \\ & & & & & f(x_{k}) \end{matrix} ) u^{∗}

for any f ∈ C(X). Since λ is surjective, F^{k}X can be regarded as a quotient space X^{k} × U(k). Therefore, for convenience, a point in F^{k}X will be written as

[x_{1}, x_{2}, ⋯, x_{k}, u]

which means λ(x_{1}, x_{2}, ⋯, x_{k}, u). With the above notation, it is easy to see that, if X is path connected and is not a single point, then any element in F^{k}X can be approximated arbitrarily well by elements in F^{k}X with distinct spectra.

2.18. If X_{1} ⊂ X is a subset, then define F^{k}X_{1} to be the subset of F^{k}X consisting of those homomorphisms ϕ ∈ Hom(C(X), M_{k}(C^{|} )) with SP(ϕ) ⊂ X_{1} as a set. Obviously, if X_{1} is open (closed resp.), then F^{k}X_{1} is open (closed resp.). If X_{1}, X_{2}, ⋯, X_{i} are disjoint subspaces of X, and k_{1}, k_{2}, ⋯, k_{i} are nonnegative integers with

k_{1} + k_{2} + ⋯ + k_{i} = k,

then define F^{(k_{1},k_{2},···,k_{i})}(X_{1}, X_{2}, ⋯, X_{i}) to be the subset of F^{k}X consisting of all ϕ with

#(SP(ϕ) ∩ X_{i}) = k_{i}

counting multiplicity. Usually when we use the above notation, we suppose that X̄_{i_{1}} ∩ X̄_{i_{2}} = ∅ if i_{1} ≠ i_{2}, where X̄_{i} is the closure of X_{i}. In this case,

F^{k}(X_{1} ∪ X_{2} ∪ ⋯ ∪ X_{i}) = ∐_{k_{1}+k_{2}+···+k_{i}=k} F^{(k_{1},k_{2},···,k_{i})}(X_{1}, X_{2}, ⋯, X_{i})

is a disjoint union of separate components.

2.19. For each i-tuple (k_{1}, k_{2}, ⋯, k_{i}) with

k_{1} + k_{2} + ⋯ + k_{i} = k,

one can define G_{(k_{1},k_{2},···,k_{i})}(C^{|} ^{k}) to be the collection of i-tuples (p_{1}, p_{2}, ⋯, p_{i}) of orthogonal projections p_{j} ∈ M_{k}(C^{|} ) with rank(p_{j}) = k_{j} (1 ≤ j ≤ i) and ∑_{j=1}^{i} p_{j} = 1 ∈ M_{k}(C^{|} ). Note that if i = 2, G_{(k_{1},k_{2})}(C^{|} ^{k}) is the ordinary complex Grassmannian manifold G_{k_{1}}(C^{|} ^{k}) = G_{k_{2}}(C^{|} ^{k}).

For each fixed i-tuple (k_{1}, k_{2}, ⋯, k_{i}), there is a locally trivial fibre bundle

F^{k_{1}}(X_{1}) × F^{k_{2}}(X_{2}) × ⋯ × F^{k_{i}}(X_{i}) ⟶ F^{(k_{1},k_{2},···,k_{i})}(X_{1}, X_{2}, ⋯, X_{i})
↓
G_{(k_{1},k_{2},···,k_{i})}(C^{|} ^{k}).

2.20. For certain purposes, it is more convenient to use CW complexes (instead of simplicial complexes). For the terminology used below, see [Wh]. Suppose that (X, A) is a relative CW complex pair. If X is path connected, then (X, A) is zero connected CW complex pair, no matter A is connected or not. In particular, (X, A) is homotopy equivalent to (X_{1}, A), where X_{1} is obtained from A by attaching finitely many cells of dimension ≥ 1 (see Theorem 2.6 of Chapter five of [Wh]). This can not be done if one only uses simplicial complex pair. (Note, we always assume our CW complexes to be finite CW complexes without saying so.) For a relative CW complex pair (X, A), define F_{A}^{m}X ⊂ F^{m}X to be the subspace consisting of those elements x ∈ F^{m}X, with

SP(x) ∩ A ≠ ∅.

(This is different from the set F^{m}A (defined in 2.18) which consists of elements x ∈ F^{m}X such that SP(x) ⊂ A.)

Lemma 2.21. Suppose that (X, A) is a relative CW complex pair. Suppose that X is obtained from A by attaching cells of dimension at least 1. It follows that the inclusion

F_{A}^{m}X ↪ F^{m}X

is m − 1 equivalent, i.e., i_{∗} : π_{j}(F_{A}^{m}X) → π_{j}(F^{m}X) is an isomorphism for any 0 ≤ j ≤ m − 2 and a surjection for j = m − 1, where i_{∗} is induced by the inclusion map. The proof of this lemma is divided into two steps.

Lemma 2.22. Lemma 2.21 is true if X is obtained from A by attaching several cells of dimension 1.

Proof: Let X = A∪e_{1}∪e_{2}∪⋯∪e_{t}, where e_{1}, e_{2}, ⋯, e_{t} are 1-cells with ∂e_{i} ⊂ A. Then F^{m}X\F_{A}^{m}X consists of those points whose spectra are contained in

e_{1}^{∘} ∪ e_{2}^{∘} ∪ ⋯ ∪ e_{t}^{∘} ,

where each e_{j}^{∘} = e_{j}\∂e_{j} is homeomorphic to (0, 1). In other words,

F^{m}X\F_{A}^{m}X = ∐_{k_{1}+k_{2}+···+k_{t}=m} F^{(k_{1},k_{2},···,k_{t})}(e_{1}^{∘} , e_{2}^{∘} , ⋯, e_{t}^{∘}).

For each fixed t-tuple (k_{1}, k_{2}, ⋯, k_{t}) with ∑ k_{i} = m, the space F^{(k_{1},k_{2},···,k_{t})}(e_{1}^{∘} , e_{2}^{∘} , ⋯, e_{t}^{∘}) is a smooth manifold. To see this, we can consider the fibre bundle

F^{k_{1}}(e_{1}^{∘}) × F^{k_{2}}(e_{2}^{∘}) × ⋯ × F^{k_{t}}(e_{t}^{∘}) ⟶ F^{(k_{1},k_{2},···,k_{t})}(e_{1}^{∘} , e_{2}^{∘} , ⋯, e_{t}^{∘})
↓
G_{(k_{1},k_{2},···,k_{t})}(C^{|} )

introduced in 2.19. Evidently, the fibre of the bundle is

F^{k_{1}}(e_{1}^{∘}) × F^{k_{2}}(e_{2}^{∘}) × ⋯ × F^{k_{t}}(e_{t}^{∘}) ≅ IR^{k_{1}^{2}} × IR^{k_{2}^{2}} × ⋯ × IR^{k_{t}^{2}}.

Note that the above fibre bundle has an obvious cross section (see [DN]). Therefore, the fibre bundle can be regarded as a smooth vector bundle with the vector space IR^{k_{1}^{2}+k_{2}^{2}+···+k_{t}^{2}} as the fibre. The zero section of the bundle has codimension

k_{1}^{2} + k_{2}^{2} + ⋯ + k_{t}^{2} ≥ k_{1} + k_{2} + ⋯ + k_{t} = m.

By a standard argument from topology, using the transversality theorem, the lemma can be proved. (See [DN, 6.3.4] for details.)

⊓ ⊔

2.23. The next step is to prove Lemma 2.21 by induction, starting with 2.22. Since the proof is a complete repetition of the proof of Theorem 6.4.2 of [DN], we omit the detail—only point out how to define the collection W (n, r) in our setting, and several small modifications. Let W (1, 0) = {A}—the set with single element: the space A. For r > 0, W (n, r) is the class of all finite CW complexes, each of which is obtained by attaching at most one cell of dimension n to a space in W (n, r − 1). Let

W (n + 1, 0) = ⋃_{r=0}^{+∞} W (n, r).

Lemma 2.22 says that F_{A}^{m}X → F^{m}X is m − 1 equivalent if X ∈ W (2, 0). In applying the argument in [DN, 6.4.2], F_{A}^{m}X is in place of F^{k}(X) and F^{m}X is in place of F^{k+1}(X). All the other parts of the proof follow from [DN]. The only thing needs mentioning is that the inclusion

F_{A}^{m}(X\α_{I}) ↪ F_{A}^{m}X

is 1-equivalent, where α_{I} is a set of finitely many points inside one of the n-cells of X, and n ≥ 2. To prove this statement, one needs to prove that any continuous map from S^{1} to F_{A}^{m}X can be perturbed to a map from S^{1} to F_{A}^{m}(X\α_{I}). To do this, he can first perturb a map to a piecewise linear map for which the image will be one dimensional. And the resulting map can be easily perturbed again to a map whose spectrum avoids any given finite set of points in any cell of dimension at least 2.

⊓ ⊔

2.24. Let X be a simplicial complex, and x_{0} ∈ X be a vertex. Define X′ to be the sub-simplicial complex consisting of all the simplices Δ ( and their faces) with Δ ∋ x_{0}. Then X′ is contractible. We also use x_{0} to denote the point in F^{m}X defined by

ϕ(f) = f(x_{0}) · 1_{m} ∈ M_{m}(C^{|} )

for each f ∈ C(X). We can easily prove the following claim: The map F^{m}X′\{x_{0}} ↪ F^{m}X′ is (m − 1) equivalent. To see this, let

A = ∪{Δ | Δ is a simplex of X′ and x_{0} ∉ Δ}.

Then A is a sub-simplicial complex of X′. A may not be connected, but (X′, A) is 0-connected. In the notation of 1.4.2,

X′ = \overline{Star(\{x_{0}\})} and A = \overline{Star(\{x_{0}\})}\Star(\{x_{0}\}).

It is obvious that A ↪ X′\{x_{0}} is a homotopy equivalence. Therefore, F_{A}^{m}X′ ↪ (F^{m}X′)\{x_{0}} is a homotopy equivalence. By Lemma 2.21, the claim holds. In particular,

π_{i}(F^{m}X′\{x_{0}}) = 0

for any 0 ≤ i ≤ m − 2, since F^{m}X′ is contractible. Equivalently,

π_{i}(F^{m}(X′\A)\{x_{0}}) = 0

for any 0 ≤ i ≤ m − 2. Note that X′\A is an open neighborhood of x_{0} ∈ X, which is the interior of X′.

2.25. Proof of Theorem 2.9. Suppose that

SP(x) = \{ \underbrace{λ_{1}, λ_{1}, ⋯, λ_{1}}_{k_{1}} , \underbrace{λ_{2}, λ_{2}, ⋯, λ_{2}}_{k_{2}} , ⋯, \underbrace{λ_{i}, λ_{i}, ⋯, λ_{i}}_{k_{i}} \},

where λ_{1}, λ_{2}, ⋯, λ_{i} ∈ X are distinct points and k_{1} + k_{2} + ⋯ + k_{i} = m. Choose mutually disjoint open sets U_{1} ∋ λ_{1}, U_{2} ∋ λ_{2}, ⋯, U_{i} ∋ λ_{i}, in X. Then there is a locally trivial fibre bundle

F^{k_{1}}(U_{1}) × F^{k_{2}}(U_{2}) × ⋯ × F^{k_{i}}(U_{i}) ⟶ F^{(k_{1},k_{2},···,k_{i})}(U_{1}, U_{2}, ⋯, U_{i})
↓
G_{(k_{1},k_{2},···,k_{i})}(C^{|} ).

Note that G_{(k_{1},k_{2},···,k_{i})}(C^{|} ) = U(m)/(U(k_{1}) × U(k_{2}) × ⋯ × U(k_{i})) is a smooth manifold of dimension t := m^{2} − ∑_{j=1}^{i} k_{j}^{2}. There is a small contractible neighborhood U_{x} ⊂ B_{ε}(x) which is homeomorphic to the space

F^{k_{1}}(X_{1}) × F^{k_{2}}(X_{2}) × ⋯ F^{k_{i}}(X_{i}) × IR^{t},

where X_{1}, X_{2}, ⋯, X_{i} are mutually disjoint open subsets of X. The space X_{i} can be chosen so that X̄_{i} is the simplicial complex X′ as in 2.24 corresponding to vertex λ_{i}, with respect to some simplicial decomposition of X. The following fact is well known in topology. Suppose that X and Y are connected CW complexes with base points x_{0}, y_{0} respectively. If X\{x_{0}} is l_{1}-connected and X\{y_{0}} is l_{2}-connected, then (X ×Y)\{x_{0}, y_{0}} is (l_{1} +l_{2} +2)- connected. Combining this fact with 2.24, we conclude that U_{x}\{x} is

∑_{j=1}^{i} k_{j} − 2 + t = m − 2 + t

connected. This ends the proof.

⊓ ⊔

3 Combinatorial Results

In this section, we will prove certain results of a combinatorial nature, for the preparation of the proof of the Decomposition Theorem—Theorem 4.1 of the next section. We will need the results in the case that certain multiplicities are general—not just equal to one. For the sake of clarity, we will first state and prove the results in the special case of multiplicity one. We will then consider the general case.

3.1. Suppose that X is a simplicial complex. Let σ denote the simplicial complex structure of X—which tells what the simplices of X are, and what the faces of each simplex of X are. In this section, we will use (X, σ) to denote the simplicial complex X with simplicial structure σ, to emphasize that we may endow the same space X with different simplicial complex structures. In this section, we will reserve the notation, σ, τ, σ_{1}, τ_{1}, ⋯, etc., for simplicial complex structures. Recall that, if Δ is a simplex, its boundary is denoted by ∂Δ. For example, if dim(Δ) = 0, i.e., Δ = {pt}, the set consisting of a single point, then ∂Δ = ∅; if dim(Δ) = 1, i.e., Δ is an interval, then ∂Δ is the set consisting of the two extreme points of the interval. Let us also consider the set Δ\∂Δ, and denote it by interior(Δ). If (X, σ) is a simplicial complex, then for any point x ∈ X, there is a unique simplex Δ such that x ∈ interior(Δ).

As usual, if each simplex of (X, σ_{1}) is a union of certain simplices of (X, σ_{2}), then we shall call σ_{2} a subdivision of σ_{1}. This is equivalent to the property that any simplex of (X, σ_{2}) is contained in a simplex of (X, σ_{1}). The notation Vertex(X, σ) (respectively, Vertex(Δ) ) will be used to denote the set of vertices of (X, σ) (or of the simplex Δ).

Definition 3.2. Let E = {1, 2, ⋯, K} be an index set. (The index set E can be any set with exactly K elements.) Let K_{1}, K_{2}, ⋯, K_{m} be non negative integers with

K_{1} + K_{2} + ⋯ + K_{m} = K.

A grouping of E of type (K_{1}, K_{2}, ⋯, K_{m}) is a collection of m mutually disjoint index sets E_{1}, E_{2}, ⋯, E_{m} with

E = E_{1} ∪ E_{2} ∪ ⋯ ∪ E_{m},

and #(E_{j}) = K_{j} for each 1 ≤ j ≤ m. (Cf. 1.5.18.) Usually, we will keep the tuple (K_{1}, K_{2}, ⋯, K_{m}) fixed and just call the collection E_{1}, E_{2}, ⋯, E_{m} a grouping of E (without mentioning the type). (Most of the time, K_{1}, K_{2}, ⋯ K_{m} will be positive integers, i.e., nonzero. But for convenience, we allow some numbers K_{i} = 0, and then the corresponding sets E_{i} should be the empty set.)

3.3. Let (X, σ) be a simplicial complex. Suppose that, associated to each vertex x ∈ X, there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E of type (K_{1}, K_{2}, ⋯, K_{m}). (In our application in the proof of Theorem 4.1, the index set E will be the spectrum of a homomorphism at the given vertex, see 1.5.13, 1.5.17–1.5.22.) Suppose that these groupings for all the vertices are chosen arbitrarily. Then, in general, for a simplex Δ with vertices x_{0}, x_{1}, ⋯, x_{n}, the intersections

⋂_{x∈Vertex(Δ)} E_{j}(x) = E_{j}(x_{0}) ∩ E_{j}(x_{1}) ∩ ⋯ ∩ E_{j}(x_{n}), j = 1, 2, ⋯, m,

may have very few elements—the sets E_{j}(x_{0}), E_{j}(x_{1}), ⋯, E_{j}(x_{n}) may be very different. The purpose of this section is to introduce a subdivision (X, τ) of (X, σ), and to associate to each new vertex of (X, τ) a grouping to make the following true: For any simplex Δ of (X, τ) (after the subdivision), for each j, the number of elements in the intersection

⋂_{x∈Vertex(Δ)} E_{j}(x)

is not much less than the number of elements in each individual set E_{j}(x) (note that #(E_{j}(x)) = K_{j} for each x); in other words, the groupings of adjacent vertices (after subdivision) should be almost as the same as each other.

First we will state the following lemma (the proof will be given in 3.15). Later on, we will need a relative version of the lemma. (See 1.5.17 to 1.5.23 for the explanations of the role of this lemma in §4. To visualize the following lemma, see 1.5.21 for the explanation of the one dimensional case.)

Lemma 3.4. Let (X, σ) be a simplicial complex consisting of a single simplex X and its faces. Suppose that associated to each x ∈ Vertex(X, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E (of type (K_{1}, K_{2}, ⋯, K_{m})). It follows that there is a subdivision (X, τ) of (X, σ), and associated to each new vertex x ∈ Vertex(X, τ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E (of type (K_{1}, K_{2}, ⋯, K_{m})), (for any old vertex of (X, σ), the grouping should not be changed), such that the following hold. For each newly introduced vertex x ∈ Vertex(X, τ),

(1)  ⋂_{y∈Vertex(X,σ)} E_{j}(y) ⊂ E_{j}(x),  j = 1, 2, ⋯, m,

and

(2)  E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ of (X, τ) (after subdivision),

(3)  #( ⋂_{x∈Vertex(Δ)} E_{j}(x)) ≥ K_{j} − \frac{n(n + 1)}{2},  j = 1, 2, ⋯, m,

where n = dim X. (When we apply this lemma in §4, the simplex X will be a simplex of a simplicial complex Y, and K_{j} >> (dim Y)^{3}; from this it follows that

#(  ⋂_{x∈Vertex(Δ)} E_{j}(x)) ≥ K_{j} − \frac{n(n + 1)}{2} >> (dim Y)^{3},  j = 1, 2, ⋯, m.)

Remark 3.5. The inclusion E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y) in the condition (2) of Lemma 3.4 is important for our application in §4 (see 1.5.19 for the explanation). We will put this inclusion into a more general context, 3.7. So we will only discuss the condition (1) in this remark. The inclusion ⋂_{y∈Vertex(X,σ)} E_{j}(y) ⊂ E_{j}(x) in the condition (1) above, will not be used in our application in §4. But taking this inclusion as a part of the conclusion will make the induction argument easier in the proof. We would like to point out that the weak version of the above lemma without requiring the inclusion in (1) will automatically imply the above stronger version. (This can be seen from the proof of Corollary 3.14 below.)

3.6. Strategy and logistics of the proof of Lemma 3.4 We shall prove it by induction. So we assume that the lemma is true for simplex of dimension at most n − 1, and prove the case that dim(X) = n. First, we introduce a new vertex, which is the barycenter of X, and introduce a model grouping E_{1}^{model}, E_{2}^{model}, ⋯, E_{m}^{model} of E for the new vertex. We shall view X as many layers similar to the boundary ∂X: ∂X × {0}, ∂X × {t_{1}}, ⋯, and the top layer ∂X × {1} is identified into a single point which is the barycenter, where t_{1}, t_{2}, ⋯, is a finite sequence of increasing numbers between 0 and 1 (the number of terms in this sequence depending in a certain sense on the distance between the giving groupings at the vertices and the model grouping at the barycenter). See the picture below.

[figure: ∂X × {0}; ∂X × {t_{1}}; ∂X × {t_{2}}]

We will introduce a subdivision of each layer ∂X ×{t_{i}} (identifying the layer as a set with ∂X and thereby endow it with a simplicial complex structure), and a grouping for each vertex on this layer. The general principle we shall follow is: the higher the layer is, the closer the groupings are to the model grouping. We should, gradually, change the groupings from each layer to the next higher layer. Let us explain it for the case dim(X) = 2 and dim(∂X) = 1. Fix a t_{i}, and suppose that we have the simplicial structure and groupings for all vertices on ∂X × {t_{i}}. Let us using the following picture to show the vertices of ∂X × {t_{i}}.

[figure: x_{1}; x_{2}; x_{3}; x_{5}; x_{4}; ∂X × {t_{i}}]

(I.e., there are five 1-dimensional simplices [x_{1}, x_{2}], [x_{2}, x_{3}], [x_{3}, x_{4}], [x_{4}, x_{5}] and [x_{5}, x_{1}].) Let us assume that the condition (3) holds for any simplex of ∂X × {t_{i}} with dim(X) replaced by dim(∂X). (We will also discuss the condition (1) below, but not the condition (2).) We shall construct simplicial structure and groupings on ∂X ×{t_{i+1}}. To begin with, let us provisionally define the simplicial structure on ∂X × {t_{i+1}} to be as the same as that on ∂X × {t_{i}}, as in the picture on the next page. Fix an element λ ∈ E_{1}^{model} such that λ ∉ ⋂_{k=1}^{5} E_{1}(x_{k}). (If such an element does not exist, then the groupings are already good for E_{1}. In other words, E_{1}(x_{k}) contains and therefore equals E_{1}^{model} for every k. Then we should go on to E_{2} or other parts.) The grouping on the vertex y_{j}, j = 1, ⋯, 5 will be taken to be either the grouping on the corresponding vertex x_{j}, if E_{1}(x_{j}) ∋ λ, or the grouping on the corresponding vertex x_{j}, with a certain element of E_{1}(x_{j})\E_{1}^{model} replaced by λ ∈ E_{1}^{model} if E_{1}(x_{j}) ∌ λ. Lemma 3.9 below tells which element should be chosen to be replaced. Of course the other part E_{t}, t > 1 of the grouping must also be slightly modified. Lemma 3.9 also guarantees that such modification exists. Subsections 3.7 and 3.8 give the definition used in 3.9. (This consideration are all in order to ensure the condition (2).)

[figure: x_{1}; x_{2}; x_{3}; x_{5}; x_{4}; y_{1}; y_{2}; y_{3}; y_{5}; y_{4}; ∂X × {t_{i}}; ∂X × {t_{i+1}}]

Now, ⋂_{k=1}^{5} E_{1}(y_{k}) contains one more element of E_{1}^{model} than ⋂_{k=1}^{5} E_{1}(x_{k}), namely, λ. So for E_{1}, the groupings on ∂X × {t_{i+1}} are (globally) closer to the model grouping than that on ∂X × {t_{i}}. But the groupings on ∂X×{t_{i+1}} may not satisfy the condition (3) with dim(X) replaced by dim(∂X), as the groupings on ∂X × {t_{i}} do . By the induction assumption, applied to each individual simplex of ∂X ×{t_{i+1}} (with the provisional simplicial structure), we can introduce a subdivision for ∂X ×{t_{i+1}} and groupings for the new vertices to make the condition (3), with dim(X) replaced by dim(∂X), hold for ∂X × {t_{i+1}}. The picture now looks like

[figure: x_{1}; x_{2}; x_{3}; x_{5}; x_{4}; y_{1}; y′_{1}; y_{2}; y′_{2}; y_{3}; y′_{5}; y_{5}; y_{4}; y′′_{3}; y′_{3}; ∂X × {t_{i}}; ∂X × {t_{i+1}}]

In this picture, y′_{1}, y′_{2}, y′_{3}, y′′_{3}, and y′_{5} are the new vertices introduced in the subdivision. (Of course this picture only shows a special case.) It goes without saying that we wish to ensure the condition (1) (and also the condition (2)) for the groupings associated to the new vertices inside each provisional simplex of ∂X × {t_{i+1}}. In the other words, when we introduce the groupings for a new vertex inside a fixed provisional simplex of ∂X × {t_{i+1}} (e.g., y′_{2} inside [y_{2}, y_{3}]), for each k we should keep the intersection of the sets E_{k} over vertices of this simplex (e.g., E_{k}(y_{2}) ∩ E_{k}(y_{3})) inside the set E_{k} for the new vertex (e.g., inside E_{k}(y′_{2})). This is the condition (1) for this provisional simplex. The condition (1) for all the individual simplices implies that after the subdivision, the intersection over the whole layer ⋂_{y∈Vertex(X×{t_{i+1}})} E_{1}(y) is equal to the intersection over the vertices of provisional simplicial structure ⋂_{k=1}^{5} E_{1}(y_{k}), and therefore still contains one more element of E_{1}^{model} than ⋂_{k=1}^{5} E_{1}(x_{k}) (namely , λ).

One may notice that the subset ∂X × [t_{i}, t_{i+1}] is not automatically a simplicial complex. We shall use Lemma 3.10 below to decompose it into a simplicial complex. Because we do not change much from the grouping of x_{j} to the grouping of y_{j} and because we make (1) true when introduce groupings for new vertices y′_{j}, y′′_{j}, etc., the groupings for any simplex inside ∂X × [t_{i}, t_{i+1}] will satisfy the condition (3) (of course with dim(X) not replaced by dim(∂X)). Finally, let us mention that, we carry out the above construction separately for E_{1}, E_{2}, etc. Once this has been done for E_{1}, the same method can be used for E_{2}. The condition (1) will guarantee that when we work on E_{2}, we will not affect the condition (3) for E_{1}, which was supposed to be already satisfied. (The details will be contained in the proof of Lemma 3.11) As we mentioned above, when we construct E_{1}(y_{j}) from E_{1}(x_{j}), we need to replace one element of E_{1}(x_{j})\E_{1}^{model} by the element λ ∈ E_{1}^{model}\E_{1}(x_{j}). If we choose an arbitrary element μ ∈ E_{1}(x_{j})\E_{1}^{model} to be replaced by λ to define E_{1}(y_{j}), then in general, E_{1}(y_{j}) may not be extended to a grouping satisfying the condition (2), in other words, there may not exist a grouping E_{1}, E_{2}, ⋯, E_{m} of E of type (K_{1}, K_{2}, ⋯, K_{m}) such that E_{1} = E_{1}(y_{j}) and

E_{k} ⊂ ⋃_{x∈Vertex(X)} E_{k}(x), k = 1, 2, ⋯, m.

So we need to give a condition to ensure that a subset E_{1} ⊂ E can be extended to a grouping satisfying condition (2). This will be discussed in 3.7 and 3.8. (See condition (∗∗) in 3.8.) The proof of Lemma 3.4 will be given in 3.7 to 3.16.

3.7. We will put the inclusion E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y) in the condition (2) of Lemma 3.4, into a more general form, as follows. (In fact, we will use this more general form in our application.) Suppose that H_{1}, H_{2}, ⋯, H_{m} are (not necessarily disjoint) subsets of E, satisfying the following condition (called Condition (∗)). For each subset I ⊂ {1, 2, ⋯, m},

#(⋃_{i∈I} H_{i}) ≥ ∑_{i∈I} K_{i} . (∗)

It follows obviously that H_{1} ∪H_{2} ∪⋯∪H_{m} = E, since #(H_{1} ∪H_{2} ∪⋯∪H_{m}) ≥ ∑_{i=1}^{m} K_{i} = #(E). From the Marriage Lemma of [HV] (or the Pairing Lemma in [Su]), the condition (∗) is a necessary and sufficient condition for the existence of a grouping E_{1}, E_{2}, ⋯, E_{m} of E of type (K_{1}, K_{2}, ⋯, K_{m}) with the condition E_{i} ⊂ H_{i}. (Recall that, the Marriage Lemma of [HV] is stated as follows. Suppose that there are two groups of K boys and K girls. Suppose that the following condition holds:

For any subset of K_{1} girls (K_{1} = 1, 2, ⋯, K), there are at least K_{1} boys, each of them knows at least one girl from this subset. Then there is a way to arrange marriage between them such that each boy marries one of the girls he knows. Our claim above is a special case of this Marriage Lemma. One can see this as follows. Suppose that the K girls are from m different clubs, and the i-th club has exactly K_{i} girls. Number the boys by 1, 2, ⋯, K. Let us define the relation consisting of a boy knowing a girl as follows. If j ∈ H_{i}, then the boy j knows all the girls in the i-th club. Otherwise, he does not know any girl in the i-th club. (Notice that the boy j could be in different H_{i}, so he could know girls from different clubs.) Obviously the condition (*) becomes the above condition in the Marriage Lemma. So if the condition (∗) holds, then there is a way to arrange the marriage as in the lemma. One can define E_{i} to be the set of boys each of whom marries a girl from the i-th club. Obviously E_{i} ⊂ H_{i}. This proves the sufficiency part of the condition. The necessary part is trivial.) If we let H_{i} = ⋃_{y∈Vertex(X,σ)} E_{i}(y), then the inclusion in (2) of Lemma 3.4 becomes E_{i}(x) ⊂ H_{i} for each x ∈ Vertex(X, σ). For any subset I ⊂ {1, 2, ⋯, m}, let

H_{I} = ⋃_{i∈I} H_{i} .

3.8. We say that a subset E_{1} ⊂ H_{1}, of K_{1} elements, satisfies Condition (∗∗) if for any I ⊂ {2, 3, ⋯, m},

(∗∗)  #(H_{I}\E_{1}) ≥ ∑_{i∈I} K_{i} .

(Caution: 1 ∉ I.) Again, from the Marriage Lemma, E_{1} ⊂ H_{1} satisfies (∗∗) if and only if E_{1} can be extended to a grouping E_{1}, E_{2}, ⋯, E_{m} of E of type (K_{1}, K_{2}, ⋯, K_{m}) such that E_{i} ⊂ H_{i}.

Lemma 3.9. Suppose that E_{1}, F_{1} (⊂ H_{1}) are two subsets satisfying (∗∗). If λ ∈ F_{1}\E_{1}, then there is a μ ∈ E_{1}\F_{1} such that

E′_{1} = (E_{1}\{μ}) ∪ {λ},

satisfies (∗∗).

Proof: Let G = E_{1} ∪ {λ}. Since E_{1} satisfies (∗∗), necessarily,

#(H_{I}\G) ≥ ∑_{i∈I} K_{i} − 1 ,

for all subsets I ⊂ {2, 3, ⋯, m}. Let H̃_{i} = H_{i}\G, i ∈ {2, 3, ⋯, m}. And let H̃_{I} = ∪_{i∈I} H̃_{i} for any I ⊂ {2, 3, ⋯, m}. The above inequality becomes #(H̃_{I}) ≥ ∑_{i∈I} K_{i} − 1 .

Let I_{0} be a minimum subset of {2, 3, ⋯, m} such that

#(H̃_{I_{0}}) = ∑_{i∈I_{0}} K_{i} − 1 .

Note that such set I_{0} exists, since if I = {2, 3, ⋯, m},

#(H̃_{I}) = ∑_{i∈I} K_{i} − 1 .

Using the fact that #(H_{I_{0}}\F_{1}) ≥ ∑_{i∈I_{0}} K_{i} , we can prove that

E_{1} ∩ H_{I_{0}} ⊄ F_{1}.

If it is not true, then G ∩ H_{I_{0}} ⊂ F_{1}, since λ ∈ F_{1} and G = E_{1} ∪ {λ}. And therefore,

#(H̃_{I_{0}}) = #(H_{I_{0}}\G) ≥ #(H_{I_{0}}\F_{1}) ≥ ∑_{i∈I_{0}} K_{i} ,

which contradicts with the above equation. Choose any element μ ∈ (E_{1} ∩ H_{I_{0}})\F_{1}; we will prove that μ is as desired in the lemma. I.e., the set

E′_{1} = (E_{1}\{μ}) ∪ {λ} = G\{μ}

satisfies (∗∗). That is, for any J ⊂ {2, 3, ⋯, m}, #(H_{J}\E′_{1}) ≥ ∑_{i∈J} K_{i} . The proof is divided into three cases.

(i) The case that J ∩ I_{0} = ∅. By the relations

H̃_{I_{0}∪J} = (H̃_{J}\H̃_{I_{0}}) ∪ H̃_{I_{0}} (disjoint union)

and

#(H̃_{I_{0}∪J}) ≥ ∑_{i∈I_{0}∪J} K_{i} − 1 ,

combined with the definition of I_{0}, one knows that

(a)  #(H̃_{J}\H̃_{I_{0}}) ≥ ∑_{i∈J} K_{i} ,

which is stronger than the condition

#(H_{J}\E′_{1}) ≥ ∑_{i∈J} K_{i} .

(ii) The case that J ⊂ I_{0}. Obviously, for J = I_{0}, we have

#(H_{I_{0}}\E′_{1}) = #(H_{I_{0}}\G) + 1 = ∑_{i∈I_{0}} K_{i} ,

since E′_{1} = G\{μ} and μ ∈ H_{I_{0}} ∩ G. So we can suppose that J \begin{matrix}⊂ \\ ≠ \end{matrix} I_{0}. By the minimality of I_{0}, we know that

#(H_{J}\G) ≥ ∑_{i∈J} K_{i} .

Therefore,

#(H_{J}\E′_{1}) ≥ ∑_{i∈J} K_{i} .

(iii) The general case. Let J_{0} = J ∩ I_{0}, and J_{1} = J\J_{0}. Then

(H_{J}\E′_{1}) ⊃ (H̃_{J_{1}}\H̃_{I_{0}}) ∪ (H_{J_{0}}\E′_{1}),

where the right hand side is a disjoint union since J_{0} ⊂ I_{0}. Evidently, this case follows from (a) above and case (ii).

⊓ ⊔

The following lemma is perhaps well known.

Lemma 3.10. Let (X, σ_{0}) be a simplicial complex and (X, σ_{1}) be a subdivision of (X, σ_{0}). It follows that there is a simplicial structure σ of X × [0, 1] such that

(1) all vertices of (X × [0, 1], σ) are on three subsets X × \{0\}, X × \{\frac{1}{2}\}, and X × {1};

(2) (X × [0, 1], σ)|_{X×{0}} = (X, σ_{0}), and (X × [0, 1], σ)|_{X×{1}} = (X, σ_{1});

(3) For a simplex Δ of (X ×[0, 1], σ), there is a simplex Δ_{0} of (X, σ_{0}) (caution: we do not use (X, σ_{1})) such that

Δ ⊂ Δ_{0} × [0, 1],

as a subset.

Proof: We prove it by induction on dim(X). If X is 0-dimensional simplicial complex which consists of finitely many points, the conclusion is obvious, since X × [0, 1] is finitely many disjoint intervals. (Note that, at this case, necessarily, (X, σ_{0}) = (X, σ_{1}).) For us to visualize the general case later on, we introduce a new vertex (x, \frac{1}{2}) ∈ X × \{\frac{1}{2}\} for each x ∈ X. That is, we divide the interval \{x\}×[0, 1] into two simplices \{x\}×[0, \frac{1}{2}] and \{x\} × [\frac{1}{2}, 1]. As the induction assumption, let us assume that the lemma is true for any n-dimensional complex. Let dim(X) = n + 1. Let X^{(n)} be the n-skeleton of (X, σ_{0}) (we use σ_{0} not σ_{1} here). By the induction assumption, there is a simplicial structure σ′ of X^{(n)} × [0, 1] such that

(1) all vertices of (X^{(n)} ×[0, 1], σ′) are on three subsets X^{(n)} ×\{0\}, X^{(n)} ×\{\frac{1}{2}\}, and X^{(n)} × {1}; (2) (X^{(n)} × [0, 1], σ′)|_{X^{(n)}×{0}} = (X^{(n)}, σ_{0}|_{X^{(n)}}), and (X^{(n)} × [0, 1], σ′)|_{X^{(n)}×{1}} = (X^{(n)}, σ_{1}|_{X^{(n)}});

(3) For an simplex Δ of (X^{(n)} × [0, 1], σ′), there is a simplex Δ_{0} of (X^{(n)}, σ_{0}) such that

Δ ⊂ Δ_{0} × [0, 1],

as a subset.

Let us introduce the simplicial structure σ on X × [0, 1] such that (X × [0, 1], σ)|_{X^{(n)}×[0,1]} = (X^{(n)} × [0, 1], σ′), (X × [0, 1], σ)|_{X×{0}} = (X, σ_{0}), and (X × [0, 1], σ)|_{X×{1}} = (X, σ_{1}).

Consider each Δ × [0, 1] for any (n + 1)-simplex Δ of (X, σ_{0}) (again, we use σ_{0} not σ_{1}). From the above, we already have the simplicial structure on the boundary

∂(Δ × [0, 1]) = (Δ × {0}) ∪ (∂Δ × [0, 1]) ∪ (Δ × {1}).

Namely, on Δ × {0}, we use σ_{0}; on ∂Δ × [0, 1], we use σ′; and on Δ × {1}, we use σ_{1}.

Let c be the barycenter of Δ, introduce a new vertex C = (c, \frac{1}{2}) ∈ X × \{\frac{1}{2}\}. The simplices of σ on Δ × [0, 1] are of the following forms.

(i) C itself is a zero dimensional simplex;

(ii) Any simplex of the boundary ∂(Δ × [0, 1]) is a simplex for σ on Δ × [0, 1]; and

(iii) For any simplex Δ′ of the boundary ∂(Δ×[0, 1]), the convex hull of Δ′∪{C} is a simplex of dimension (dim(Δ′) + 1) for σ on Δ × [0, 1].

Define such simplicial structure for each (n + 1)-simplex separately, and put them together give rise to a simplicial structure of X × [0, 1], which obviously satisfies the conditions (1), (2), and (3).

(Note that the simplicial structure on ∂Δ × [0, 1] is as the same as σ′, therefore the simplicial structure on Δ × [0, 1] and on Δ_{1} × [0, 1] for different (n + 1)- dimensional simplices Δ and Δ_{1} are compatible on the intersection (Δ ∩ Δ_{1}) × [0, 1].)

The following pictures may help the reader to visualize the construction. They are pictures only for the case n = 0, dim(Δ) = 1, and dim(Δ × [0, 1]) = 2.

Suppose the simplicial structure for the boundary ∂(Δ × [0, 1]) is as follows. (The dots represent vertices.)

[figure: Δ × {0}; Δ × {1}]

Then the simplicial structure on Δ × [0, 1] will be described by the following picture.

[figure: Δ × \{0\}; Δ × \{1\}; C; Δ × \{\frac{1}{2}\}]
⊓ ⊔

The following lemma presents the main technical step of this section.

Lemma 3.11. Suppose that {H_{1}, H_{2}, ⋯, H_{m}} satisfies the condition (∗). Suppose that (X, σ) is a simplicial complex consisting of a single simplex Δ_{0} and its faces. Let (Y, σ) = (∂Δ_{0}, σ), and (Y, τ) be a subdivision of (Y, σ). Suppose that it is assigned, for each vertex x ∈ (Y, τ), a set E_{1}(x) ⊂ H_{1} which satisfies the condition (∗∗). Furthermore, suppose that for any simplex Δ of (Y, τ),

# ( ⋂_{y∈Vertex(Δ)} E_{1}(y)) ≥ K_{1} − \frac{dim Y (dim Y + 1)}{2} .

It follows that there are a subdivision (X, τ̃) of (X, σ) and an assignment, for each vertex x ∈ Vertex(X, τ̃), a set E_{1}(x) ⊂ H_{1}, satisfying condition (∗∗), with the following conditions.

(1) (X, τ̃)|_{Y} = (Y, τ), and for each vertex y ∈ Vertex(Y, τ), the assignment E_{1}(y) is as same as the original one.

(2) For any x ∈ (X, τ̃),

E_{1}(x) ⊃ ⋂_{y∈Vertex(Y,τ)} E_{1}(y) .

(3) For any simplex Δ of (X, τ̃),

# ( ⋂_{x∈Vertex(Δ)} E_{1}(x)) ≥ K_{1} − \frac{dim X(dim X + 1)}{2} .

Proof: The Lemma is proved by induction on the dimension of the simplex. If dim(Δ_{0}) = 0, then Δ_{0} = {pt}, a set of single point, and ∂Δ_{0} = ∅. Obviously, the lemma holds by choosing any E_{1}(pt) ⊆ H_{1} of K_{1} element to satisfy (∗∗). Let us prove the 1-dimensional case. Logically, this part could be skipped. But the proof of this case will be easier to visualize which can be used to understand the general case. Suppose that dim(Δ_{0}) = 1. Δ_{0} is a line segment [0, 1]. Divide [0, 1] into several subintervals by

0 = t_{0}^{0} < t_{1}^{0} < t_{2}^{0} < ⋯ < t_{a−1}^{0} < t_{a}^{0} = \frac{1}{2} = t_{a}^{1} < t_{a−1}^{1} < ⋯ < t_{2}^{1} < t_{1}^{1} < t_{0}^{1} = 1.

(The natural number a is to be determined later.) The points {t_{j}^{i}}_{i=0,1;j=1,2···,a} will be the new vertices of (Δ_{0}, τ̃). (Note that t_{a}^{0} is the same vertex as t_{a}^{1}.) Choose a model E_{1}^{model} ⊂ H_{1} to satisfy (∗∗) and

E_{1}^{model} ⊃ E_{1}(t_{0}^{0}) ∩ E_{1}(t_{0}^{1}).

In fact, one can choose E_{1}^{model} to be either E_{1}(t_{0}^{0}) or E_{0}(t_{0}^{1}). (Note that t_{0}^{0} = 0 and t_{0}^{1} = 1 are vertices of (Δ_{0}, σ).) Let G = E_{1}(t_{0}^{0}) ∩ E_{1}(t_{0}^{1}) ∩ E_{1}^{model}. Without loss of generality, we can assume that there is λ ∈ E_{1}^{model}\G. Otherwise, E_{1}^{model} = G = E_{1}(t_{0}^{0}) = E_{1}(t_{0}^{1}), and the conclusion already holds before introducing any subdivision. By Lemma 3.9, if λ ∉ E_{1}(t_{0}^{i}) (i = 0, 1), then there is a μ ∈ E_{1}(t_{0}^{i})\E_{1}^{model} such that E_{1}(t_{0}^{i}) ∪ {λ}\{μ} satisfies (∗∗). Define

E_{1}(t_{1}^{i}) = \{ \begin{matrix}E_{1}(t_{0}^{i}) & if λ ∈ E_{1}(t_{0}^{i}) \\ (E_{1}(t_{0}^{i}) ∪ \{λ\})\\{μ\} & if λ ∉ E_{1}(t_{0}^{i}) . \end{matrix}

Then E_{1}(t_{1}^{i}) ⊃ G ∪ {λ}. Therefore,

E_{1}(t_{1}^{0}) ∩ E_{1}(t_{1}^{1}) ∩ E_{1}^{model} \begin{matrix}⊃ \\ ≠ \end{matrix} E_{1}(t_{0}^{0}) ∩ E_{1}(t_{0}^{1}) ∩ E_{1}^{model}.

Suppose that we already have the definitions of E_{1}(t_{i}^{0}) and E_{1}(t_{i}^{1}), we can define E_{1}(t_{i+1}^{0}) and E_{1}(t_{i+1}^{1}) exactly the same as above (i in place of 0, and i + 1 in place of 1), and obtain

E_{1}(t_{i+1}^{0}) ∩ E_{1}(t_{i+1}^{1}) ∩ E_{1}^{model} \begin{matrix}⊃ \\ ≠ \end{matrix} E_{1}(t_{i}^{0}) ∩ E_{1}(t_{i}^{1}) ∩ E_{1}^{model}.

Carrying out this procedure for at most finitely many times, we will reach E_{1}(t_{a−1}^{0}) ∩ E_{1}(t_{a−1}^{1}) ∩ E_{1}^{model} = E_{1}^{model}. Then define E_{1}(t_{a}^{i}) = E_{1}^{model}. (Note that t_{a}^{1} = t_{a}^{0} = \frac{1}{2}.) For i = 0, 1; j = 0, 1, 2, ⋯, a − 1,

#(E_{1}(t_{j}^{i}) ∩ E_{1}(t_{j+1}^{i})) ≥ K_{1} − 1 = K_{1} − \frac{dim(Δ_{0})(dim(Δ_{0}) + 1)}{2},

since we take out at most one point from E_{1}(t_{j}^{i}) to define E_{1}(t_{j+1}^{i}). This proves that the lemma holds for n = 1. (Let us point out that for one dimensional case, the proof could be simpler. We choose the above proof to present some idea for the general case below.) Suppose that the lemma is true for any simplex of dimension ≤ n − 1. We will prove it for dim(X) = n. (One should compare to the explanation in 3.6.) Step 1. Identify Δ_{0} with ∂Δ_{0} ×[0, 1]/∂Δ_{0} ×{1}. Regard ∂Δ_{0} as ∂Δ_{0} ×{0} ⊂ ∂Δ_{0} × [0, 1]. Note that ∂Δ_{0} × {1} is identified as a single point which is the center of Δ_{0}, and is NOT a vertex of (Δ_{0}, σ). Choose 0 = t_{0} < t_{1} < ⋯ < t_{a} = 1. (The natural number a is to be determined later.) We will first introduce some new vertices (for the subdivision (X, τ̃)) on ∂Δ_{0} × {t_{1}}, ∂Δ_{0} × {t_{2}}, ⋯, ∂Δ_{0} × {t_{a}}, and define E_{1} for those vertices. Later on (in Step 4), we will consider each ∂Δ × [t_{i}, t_{i+1}] to be X × [0, 1] in Lemma 3.10, and introduce new vertices on ∂Δ×\{\frac{t_{i}+t_{i+1}}{2} \} (in place of X×\{\frac{1}{2}\}). (We need to do this, because ∂Δ × [t_{i}, t_{i+1}] is not automatically a simplicial complex.)

Choose a model E_{1}^{model} ⊂ H_{1} to satisfy (∗∗). We also require that

E_{1}^{model} ⊃ ⋂_{y∈Vertex(Y,τ)} E_{1}(y) .

(One can choose E_{1}^{model} to be E_{1}(y) for any vertex y ∈ Vertex(Y, τ).) Define E_{1}{(x, 1)} = E_{1}^{model}. (Note that {(x, 1)} ⊂ ∂Δ_{0} × [0, 1] is identified to a single point, the center of Δ_{0}.) The construction will be carried out in Step 2, 3 and 4. The procedure can be outlined as follows. If we already have the construction of simplicial structure τ̃ for ∂Δ_{0}×{t_{i−1}} and the definition of E_{1} on all vertices in Vertex(∂Δ_{0}×{t_{i−1}}), then, to define the simplicial structure on ∂Δ_{0} × [t_{i−1}, t_{i}] (in particular, to introduce vertices on ∂Δ_{0} × {t_{i}}), and to define E_{1} on the newly introduced vertices (on ∂Δ_{0} × {t_{i}}), we will only use the simplicial structure and the definition of E_{1} on ∂Δ_{0} × {t_{i−1}}. In this procedure, if there is a vertex x ∈ Vertex(∂Δ_{0} × {t_{i−1}}, τ̃) such that

E_{1}(x) ≠ E_{1}^{model} ,

then we will require that

(a) ⋂_{x∈Vertex(∂Δ_{0}×\{t_{i}\},τ̃)} E_{1}(x) ∩ E_{1}^{model} \begin{matrix}⊃ \\ ≠ \end{matrix} ⋂_{x∈Vertex(∂Δ_{0}×\{t_{i−1}\},τ̃)} E_{1}(x) ∩ E_{1}^{model} .

(That is, the sets E_{1}’s on ∂Δ_{0} × {t_{i}} are globally closer to E_{1}^{model} than those on ∂Δ_{0} × {t_{i−1}}.) Finally, within finitely many steps, we will reach that, for certain i − 1, and for all vertices x ∈ Vertex(∂Δ_{0} × {t_{i−1}}, τ̃),

E_{1}(x) = E_{1}^{model} .

Then we choose t_{i} = t_{a} = 1, and choose any simplicial structure on ∂Δ_{0} × [t_{a−1}, 1]/∂Δ_{0} × {1} with vertex set to be Vertex(∂Δ_{0} × {t_{a−1}}) ∪ ∂Δ_{0} × {1}. Recall that the set ∂Δ_{0}×{1} is identified as a single point with E_{1}(∂Δ_{0}×{1}) = E_{1}^{model}.) Furthermore, in this procedure, we not only make (3) true for any simplex in ∂Δ_{0} × [t_{i−1}, t_{i}], but also make the following stronger statement true for any simplex Δ lies on ∂Δ_{0} × {t_{i}}:

(b)  # (  ⋂_{y∈Vertex(Δ)} E_{1}(y)) ≥ K_{1} − \frac{(n − 1)n}{2} .

(Note that n − 1 = dim(∂Δ_{0} × {t_{i}}) = dim(∂Δ_{0}).) This condition has to be satisfied for the construction of the next step by induction. Step 2. We will do all the above construction only for ∂Δ_{0} × [t_{0}, t_{1}]. For the other part of the construction, one uses induction argument with aid of (b) (i.e., let t_{i−1} play the role of t_{0}, and t_{i} play the role of t_{1}.)

Let {(y_{1}, t_{0}), (y_{2}, t_{0}), ⋯, (y_{p}, t_{0})} be the vertices of ∂Δ_{0} ×{t_{0}} = Y. There is a simplicial complex structure on ∂Δ_{0}×{t_{1}}, which is exactly the same as that of (∂Δ_{0} ×{t_{0}}, τ̃), since both ∂Δ_{0} ×{t_{1}} and ∂Δ_{0} ×{t_{0}} can be regarded as ∂Δ_{0}. We call such simplicial complex τ̃_{pre}. Therefore, each point (y_{i}, t_{1}) (1 ≤ i ≤ p) is a vertex of (∂Δ_{0} × {t_{1}}, τ̃_{pre}). We will introduce more vertices later. Let G = E_{1}(y_{1}, t_{0}) ∩ E_{1}(y_{2}, t_{0}) ∩ ⋯ ∩ E_{1}(y_{p}, t_{0}) ∩ E_{1}^{model}. If G = E_{1}^{model}, then E_{1}(y_{i}, t_{0}) = E_{1}^{model} for each 1 ≤ i ≤ p and the construction is done. So we assume

G ≠ E_{1}^{model}.

Choose λ ∈ E_{1}^{model}\G. When we define E_{1}(x) for any vertex x ∈ ∂Δ_{0} × {t_{1}}, it is always required that

E_{1}(x) ⊃ G ∪ {λ}.

Therefore, (a) holds for the pair {t_{0}, t_{1}}. For each point (y_{i}, t_{0}), if λ ∉ E_{1}(y_{i}, t_{0}), by Lemma 3.9, there is μ ∈ E_{1}(y_{i}, t_{0})\E_{1}^{model} such that (E_{1}(y_{i}, t_{0}) ∪ {λ})\{μ} satisfies (∗∗). Define

E_{1}(y_{i}, t_{1}) = \{ \begin{matrix}E_{1}(y_{i}, t_{0}) & if λ ∈ E_{1}(y_{i}, t_{0}) \\ (E_{1}(y_{i}, t_{0}) ∪ \{λ\})\\{μ\} & if λ ∉ E_{1}(y_{i}, t_{0}) . \end{matrix}

In this way, obviously, E_{1}(x) ⊃ G ∪ {λ} for each vertex x = (y_{i}, t_{1}) ∈ Vertex(∂Δ_{0} × {t_{1}}, τ̃_{pre}). Step 3. Note that the definition of E_{1} on Vertex(∂Δ_{0} × {t_{1}}, τ̃_{pre}) may not satisfies (b). Therefore we can not use the simplicial structure τ̃_{pre} and the definition of E_{1} on Vertex(∂Δ_{0} × {t_{1}}, τ̃_{pre}) to construct simplicial structure and the definition of E_{1} for ∂Δ_{0} × {t_{2}}. We need to introduce a subdivision for (∂Δ_{0} × {t_{1}}, τ̃_{pre}) and the definitions of E_{1} for new vertices to make (b) true. (This step is not needed in the one dimensional case, since for any zero dimensional simplex (which is a point), (b) automatically holds.) Apply the induction assumption to each simplex of (∂Δ_{0} × {t_{1}}, τ̃_{pre}) with the above definition of E_{1} on Vertex(∂Δ_{0} × {t_{1}}, τ̃_{pre}), from the simplices of the lowest dimension ( dimension 1) to the simplices of the highest dimension (dimension n − 1). (Note that each such simplex has dimension at most n − 1.) One should begin with each 1-simplex (with boundary being two points — two 0-simplices), then each 2-simplex, and so on. First, let e be any 1-simplex of (∂Δ_{0} × {t_{1}}, τ̃_{pre}) with boundary ∂e = {v_{0}, v_{1}}. Obviously, the condition of Lemma 3.11 automatically holds for simplex e in place of Δ_{0} and ∂e in place of ∂Δ_{0}, since ∂e is zero-dimensional. By the induction assumption, there is a subdivision (e, τ̃) of (e, τ̃_{pre}) and the definition of E_{1} for each vertex of (e, τ̃) such that

(1) The definition of E_{1} on the original vertices {v_{0}, v_{1}} are the same as before.

(2) For any x ∈ Vertex(e, τ̃),

E_{1}(x) ⊃ E_{1}(v_{0}) ∩ E_{1}(v_{1}).

(3) For any simplex e′ of (e, τ̃) (a line segment of e)

⋂_{x∈Vertex(e′)} E_{1}(x) ≥ K_{1} − \frac{dim(e)(dim(e) + 1)}{2}.

After we have done the above procedure for each 1-simplex, we can do it for each 2-simplex, since we already have simplicial structure and the definition of E_{1} for the boundary of any 2-simplex as required in the condition of Lemma 3.11. Going through this way, finally, one obtains a subdivision (∂Δ_{0} × {t_{1}}, τ̃) of (∂Δ_{0} × {t_{1}}, τ̃_{pre}) and the definition of E_{1} for each newly introduced vertex, such that the following two statements hold.

1. For each old simplex Δ of (∂Δ_{0} × {t_{1}}, τ̃_{pre}) and any new vertex x ∈ Δ,

(c)  E_{1}(x) ⊃ ⋂_{y∈Vertex(Δ,τ̃_{pre})} E_{1}(y) .

2. If Δ is a simplex of (∂Δ_{0} × {t_{1}}, τ̃), then

# ( ⋂_{y∈Vertex(Δ)} E_{1}(y)) ≥ K_{1} − \frac{dim Y (dim Y + 1)}{2} = K_{1} − \frac{(n − 1)n}{2} .

(This is the requirement (b) in Step 1.) The first statement is the induction assumption of validity of (2) and the second statement is the induction assumption of validity of (3). Step 4. In this step, we will apply Lemma 3.10 to define the simplicial structure τ̃ on ∂Δ_{0} × [t_{0}, t_{1}] and the definitions of E_{1} on all vertices. Note that we already have simplicial structure τ̃ on ∂Δ_{0} × {t_{0}} and on ∂Δ_{0} × {t_{1}}. Furthermore, τ̃|_{∂Δ_{0}×{t_{1}}} is a subdivision of τ̃|_{∂Δ_{0}×{t_{0}}} if we regard both ∂Δ_{0} × {t_{0}} and ∂Δ_{0} ×{t_{1}} as ∂Δ_{0}. Apply Lemma 3.10 (with ∂Δ_{0} in place of X) to obtain the simplicial structure on ∂Δ_{0}×[t_{0}, t_{1}] (we only need to introduce new vertices on ∂Δ_{0} × \{\frac{t_{0}+t_{1}}{2} \}). For each new vertex (u, \frac{t_{0}+t_{1}}{2} ) ∈ ∂Δ_{0} × \{\frac{t_{0}+t_{1}}{2} \}, consider (u, t_{0}) ∈ ∂Δ_{0} × \{t_{0}\}. From 3.1, there is a unique simplex Δ of (∂Δ_{0} × {t_{0}}, τ̃) such that (u, t_{0}) ∈ interior(Δ). Choose any vertex x of Δ and define E_{1}(u, \frac{t_{0}+t_{1}}{2} ) = E_{1}(x). So we have the simplicial structure τ̃ on ∂Δ_{0} × [t_{0}, t_{1}] and the definition of E_{1}(x) for each x ∈ Vertex(∂Δ_{0} × [t_{0}, t_{1}]). We need to verify the condition (3). Let Δ be any simplex of (∂Δ_{0} × {t_{0}}, τ̃) with vertices {(u_{0}, t_{0}), (u_{1}, t_{0}), ⋯, (u_{i}, t_{0})}. Then

#(E_{1}(u_{0}, t_{0}) ∩ E_{1}(u_{1}, t_{0}) ∩ ⋯ ∩ E_{1}(u_{i}, t_{0})) ≥ K_{1} − \frac{dim Y (dim Y + 1)}{2} .

Documenta Mathematica 7 (2002) 255–461

Let G_{1} = E_{1}(u_{0}, t_{0}) ∩ E_{1}(u_{1}, t_{0}) ∩ ⋯ ∩ E_{1}(u_{i}, t_{0}). From the above definition of E_{1} for vertices of ∂Δ_{0} × \{\frac{t_{0}+t_{1}}{2} \}, we know that if (u, t_{0}) ∈ Δ and (u, \frac{t_{0}+t_{1}}{2} ) ∈ Vertex(∂Δ_{0} × [0, 1], τ̃), then

(d)  E_{1}(u, \frac{t_{0} + t_{1}}{2} ) ⊃ G_{1}.

Since each E_{1}(u_{j}, t_{1}) is either E_{1}(u_{j}, t_{0}) or is obtained by replacing one element of E_{1}(u_{j}, t_{0}) by λ, we have

(e)  #(G_{1} ∩ E_{1}(u_{0}, t_{1}) ∩ E_{1}(u_{1}, t_{1}) ∩ ⋯ ∩ E_{1}(u_{i}, t_{1})) ; \begin{matrix}≥ K_{1} − \frac{dim Y (dim Y +1)}{2} − (i + 1) \\ ≥ K_{1} − \frac{(n−1)n}{2} − n \\ = K_{1} − \frac{n(n+1)}{2} . \end{matrix}

(Note that there are i + 1 (≤ n) sets of {E_{1}(u_{j}, t_{1})}_{j=0}^{i}, and, therefore, at most i + 1 points were taken out from G_{1}.) Recall that τ̃ on Δ_{0} × {t_{1}} is the subdivision of τ̃_{pre}. By (c) of Step 3, we have

⋂_{x∈Vertex(Δ×{t_{1}},τ̃)} E_{1}(x) ⊃ ⋂_{y∈Vertex(Δ×{t_{1}},τ̃_{pre})} E_{1}(y)

= E_{1}(u_{0}, t_{1}) ∩ E_{1}(u_{1}, t_{1}) ∩ ⋯ ∩ E_{1}(u_{i}, t_{1}).

(Note that (c) implies that the above “⊃” holds if the left hand side of “⊃” is replaced by E_{1}(x) for any x ∈ Vertex(Δ × {t_{1}}, τ̃), so it also holds for the intersection of these E_{1}(x). In fact, the above “⊃” can be replaced by “=”.) Then combining it with (d), we have

⋂_{x∈Vertex(Δ×[t_{0},t_{1}],τ̃)} E_{1}(x) = G_{1} ∩ E_{1}(u_{0}, t_{1}) ∩ E_{1}(u_{1}, t_{1}) ∩ ⋯ ∩ E_{1}(u_{i}, t_{1})

which has at least K_{1} − \frac{n(n+1)}{2} elements by (e). Combining this fact with (3) of Lemma 3.10, we know that the desired condition (3) holds for any simplex of (∂Δ_{0} × [t_{0}, t_{1}], τ̃). Evidently, (2) holds from the construction. Since (b) holds for ∂Δ_{0} × {t_{1}}, one can continue this procedure. This ends the proof.

⊓ ⊔

Corollary 3.12. Suppose that {H_{1}, H_{2}, ⋯, H_{m}} satisfies the condition (∗). Suppose that (X, σ) is a simplicial complex consisting of a single simplex and its faces. Suppose that there is assigned, for each vertex x ∈ (X, σ), a set E_{1}(x) ⊂ H_{1} which satisfies the condition (∗∗). It follows that there are a subdivision (X, τ) of (X, σ) and an assignment, for each new vertex x ∈ Vertex(X, τ), a set E_{1}(x) ⊂ H_{1}, satisfying condition (∗∗), with the following conditions. (The definition of E_{1} for the old vertex should not be changed.)

(1) For any x ∈ Vertex(X, τ),

E_{1}(x) ⊃ ⋂_{y∈Vertex(X,σ)} E_{1}(y) .

(2) For any simplex Δ of (X, τ),

# ( ⋂_{x∈Vertex(Δ)} E_{1}(x)) ≥ K_{1} − \frac{dim X(dim X + 1)}{2} .

Proof: To prove this corollary, one needs to apply Lemma 3.11 to simplices from the lowest dimension (e.g. dimension one simplex whose boundary consists two vertices of (X, σ)) to the highest dimension (e.g the simplex X itself with boundary ∂X). Each time, we only work on a single simplex Δ of (X, σ). And when we work on Δ, we should assume that we already have the subdivision and the definition of E_{1} on the boundary ∂Δ to satisfy the condition in Lemma 3.11 with dim(∂Δ) in place of dim(Y).

⊓ ⊔

Corollary 3.13. Let (X, σ) be a simplicial complex consisting of a single simplex X and all its faces. Suppose that associated to each x ∈ Vertex(X, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E. It follows that there is a subdivision (X, τ) of (X, σ), and associated to each new vertex x ∈ Vertex(Δ, τ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E (for any old vertex of (Δ, σ), the grouping should not be changed), such that the following hold. For each newly introduced vertex x ∈ Vertex(X, τ),

(2)  E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ of (X, τ) (after subdivision),

(3)  #( ⋂_{x∈Vertex(Δ)} E_{1}(x)) ≥ K_{1} − \frac{n(n + 1)}{2}

where n = dim X. (In this corollary, we do not require the condition (1) in Lemma 3.4. This will be done in the next corollary.)

Proof: Set ⋃_{y∈Vertex(X,σ)} E_{j}(y) := H_{j}, j = 1, 2, ⋯, m. Then H_{1}, H_{2}, ⋯, H_{m} satisfy condition (∗), and for each x ∈ Vertex(X, σ), E_{1}(x) ⊂ H_{1} satisfies (∗∗). Applying Corollary 3.12, we obtain the subdivision (X, τ) and the definition of E_{1}(x), for each new vertex, to satisfy condition (∗∗), and (1) and (2) in the Corollary 3.12. For each new vertex x, since E_{1}(x) satisfies (∗∗), we can extend it to a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) such that E_{i}(x) ⊂ H_{i}. Therefore this grouping satisfy the condition (2) of our corollary. The condition (3) follows from the condition (2) of Corollary 3.12. Thus the corollary is proved.

⊓ ⊔

Corollary 3.14. Let (X, σ) be a simplicial complex consisting of a single simplex X and all its faces. Suppose that associated to each x ∈ Vertex(X, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E. It follows that there is a subdivision (X, τ) of (X, σ), and associated to each new vertex x ∈ Vertex(Δ, τ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E (for any old vertex of (Δ, σ), the grouping should not be changed), such that the following hold. For each newly introduced vertex x ∈ Vertex(X, τ),

(1)  ⋂_{y∈Vertex(X,σ)} E_{j}(y) ⊂ E_{j}(x),  j = 1, 2, ⋯, m,

and

(2)  E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ of (X, τ) (after subdivision),

(3)  #( ⋂_{x∈Vertex(Δ)} E_{1}(x)) ≥ K_{1} − \frac{n(n + 1)}{2}.

where n = dim X. (Comparing this Corollary to Lemma 3.4, the only difference is that we require (3) holds only for E_{1} in the corollary.)

Proof: The only difference between this corollary and Corollary 3.13 is that we require condition (1) holds. To make (1) hold, we need to do the following. Reserve all the subsets ⋂_{y∈Vertex(X,σ)} E_{j}(y), j = 1, 2, ⋯, m, which are supposed to be in E_{j}(x) (if we want the condition (1) to hold), for any newly introduced vertex x; group the rest of the elements of E (using Corollary 3.13); and finally put ⋂_{y∈Vertex(X,σ)} E_{j}(y) into each E_{j}(x). The details are as follows. Set ⋂_{y∈Vertex(X,σ)} E_{j}(y) := D_{j}, j = 1, 2, ⋯, m. Then D_{j}, j = 1, 2, ⋯, m are mutually disjoint. To see this, we fix a y ∈ Vertex(X, σ), and notice that D_{j} ⊂ E_{j}(y), and E_{j}(y), j = 1, 2, ⋯, m are mutually disjoint, from the definition of grouping. Similarly, if j_{1} ≠ j_{2}, then D_{j_{1}} ∩ E_{j_{2}}(y) = ∅, for any y ∈ Vertex(X, σ). Consider E′ = E\(∪_{j}D_{j}) and the m-tuple

(K′_{1}, K′_{2}, ⋯, K′_{m}) = (K_{1} − #(D_{1}), K_{2} − #(D_{2}), ⋯, K_{m} − #(D_{m})).

For any y ∈ Vertex(X, σ), the grouping E_{1}(y), E_{2}(y), ⋯, E_{m}(y) of E of type (K_{1}, K_{2}, ⋯, K_{m}) induces a grouping E′_{1}(y), E′_{2}(y), ⋯, E′_{m}(y) of E′ of type (K′_{1}, K′_{2}, ⋯, K′_{m}), by setting E′_{j}(y) = E_{j}(y)\D_{j}. Apply Corollary 3.13 to the simplex (X, σ) and those groupings of E′, to obtain a subdivision (X, τ) and groupings E′_{1}(x), E′_{2}(x), ⋯, E′_{m}(x) of E′ for all newly introduced vertices x ∈ Vertex(X, τ) such that the following hold. For each newly introduced vertex x ∈ Vertex(X, τ),

(2′)  E′_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E′_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ of (X, τ) (after subdivision),

(3′)  #(  ⋂_{x∈Vertex(Δ)} E′_{1}(x)) ≥ K′_{1} − \frac{n(n + 1)}{2},

where n = dim X. Finally, let E_{j}(x) = E′_{j} ∪ D_{j} for any x ∈ Vertex(X, τ). Then the desired condition (1) of the corollary means D_{j} ⊂ E_{j}(x), which is true from the definition. Also the conditions (2) and (3) of the corollary follows from (2′) and (3′).

⊓ ⊔

Corollary 3.15. Suppose that (X, σ) is a simplicial complex. Suppose that for each vertex x ∈ Vertex(X, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E. It follows that there is a subdivision (X, τ) of (X, σ), and there is an extension of the definition of the groupings of E for Vertex(X, σ) to the groupings of E for Vertex(X, τ) ⊃ Vertex(X, σ) such that the following properties hold. For each newly introduced vertex x ∈ Vertex(X, τ), if x ∈ Δ, where Δ is a simplex of (X, σ)(before subdivision), then

(1)  ⋂_{y∈Vertex(Δ,σ)} E_{j}(y) ⊂ E_{j}(x),  j = 1, 2, ⋯, m,

and

(2)  E_{j}(x) ⊂ ⋃_{y∈Vertex(Δ,σ)} E_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ_{1} of (X, τ) (after subdivision),

(3)  #(  ⋂_{x∈Vertex(Δ_{1})} E_{1}(x)) ≥ K_{1} − \frac{n(n + 1)}{2},

where n = dim X. (The above (1) and (2) imply that for any x ∈ Vertex(X, τ),

⋂_{y∈Vertex(X,σ)} E_{j}(y) ⊂ E_{j}(x) ⊂ ⋃_{y∈Vertex(X,σ)} E_{j}(y),  j = 1, 2, ⋯, m.)

Proof: The proof is exactly the same as that of Corollaries 3.13 and 3.14. In fact, in the proof of Corollary 3.13, we were working simplex by simplex from the lowest dimension to the highest dimension. As same as Corollary 3.13, when we work on simplex Δ, we should suppose that, we have already done with ∂Δ. The only difference is the following. We should choose the sets H_{i}, D_{i} differently according to the simplex we are working on. For simplex Δ, choose H_{i} = ⋃_{y∈Vertex(Δ)} E_{i}(y), D_{i} = ⋂_{y∈Vertex(Δ)} E_{i}(y), i = 1, 2 ⋯, m.

⊓ ⊔

Lemma 3.4 is a special case of the following theorem.

Theorem 3.16. Suppose that (X, σ) is a simplicial complex. Suppose that for each vertex x ∈ Vertex(X, σ), there is a grouping E_{1}(x), E_{2}(x), ⋯, E_{m}(x) of E. It follows that there is a subdivision (X, τ) of (X, σ), and there is an extension of the definition of the groupings of E for Vertex(X, σ) to the groupings of E for Vertex(X, τ) ⊃ Vertex(X, σ) such that the following properties hold. For each newly introduced vertex x ∈ Vertex(X, τ), if x ∈ Δ, where Δ is a simplex of (X, σ) (before subdivision), then

(1)  ⋂_{y∈Vertex(Δ,σ)} E_{j}(y) ⊂ E_{j}(x),  j = 1, 2, ⋯, m,

and

(2)  E_{j}(x) ⊂ ⋃_{y∈Vertex(Δ,σ)} E_{j}(y),  j = 1, 2, ⋯, m.

For any simplex Δ_{1} of (X, τ) (after subdivision),

(3)  #(  ⋂_{x∈Vertex(Δ_{1})} E_{j}(x)) ≥ K_{j} − \frac{n(n + 1)}{2}, j = 1, 2, ⋯, m,

where n = dim X.

Proof: We will apply Corollary 3.15 to prove our theorem. First we can apply Corollary 3.15 to E_{1} and (X, σ) to make the condition (3) of the theorem hold for E_{1} and any simplex of the subdivision, and also the conditions (1) and (2) of the theorem hold. We call the simplicial structure after this step, τ_{1}. Then we apply Corollary 3.15 to E_{2} (in place of E_{1} ) and (X, τ_{1}) (in place of (X, σ)). We call this new subdivision τ_{2}. Now (3) for E_{2} holds for any simplex of new subdivision τ_{2}. Furthermore (1) and (2) of Corollary 3.15 hold for (X, τ_{1}) as the simplicial structure before subdivision (i.e., in place of (X, σ)) and (X, τ_{2}) as the subdivision (i.e., in place of (X, τ)). The important point is, (3) for E_{1} holds for any simplex Δ_{2} of (X, τ_{2}), because (3) for E_{1} holds for the simplex Δ_{1} of (X, τ_{1}) which supports Δ_{2} (i.e., Δ_{1} ⊃ Δ_{2}), and because (1) holds for τ_{1} (in place of σ) and τ_{2} (in place of τ). So, now (3) holds for both E_{1} and E_{2}. (It is also obvious that (1) for σ and τ_{2} ( in place of τ) follows from (1) for σ and τ_{1} (in place of τ), together with (1) for τ_{1} (in place of σ) and τ_{2} (in place of τ). The same thing also holds for (2).) Repeating this procedure, we can define τ_{3}, τ_{4}, and so on, until τ_{m}. Then (1), (2), (3) hold for σ and τ_{m} and any E_{j}, j = 1, 2, ⋯, m. Let τ = τ_{m}.

⊓ ⊔

Remark 3.17. Let us remark that, in the proof of Lemma 3.11 when we construct the sets E_{1}, simplex by simplex for (X, σ), it is impossible to obtain

# ( ⋂_{x∈Vertex(Δ′)} E_{1}(x)) ≥ K_{1} − \frac{dim(Δ′)(dim(Δ′) + 1)}{2} ,

for each simplex Δ′ of subdivision (X, τ) of (X, σ). (Explained below.) But from the proof of Corollary 3.12, we can make the following hold,

# ( ⋂_{x∈Vertex(Δ′)} E_{1}(x)) ≥ K_{1} − \frac{dim(Δ)(dim(Δ) + 1)}{2} ,

where Δ is any simplex of (X, σ) which support Δ′ (i.e., Δ′ ⊂ Δ as spaces). In other words,

# ( ⋂_{x∈Vertex(Δ′)} E_{1}(x)) ≥ K_{1} − \frac{l(l + 1)}{2} ,

if Δ′ is a subspace of l-skeleton X^{(l)} of (X, σ). In the induction construction from dimension not larger than n−1 to dimension n (see the proof of Lemma 3.11), in particular, from (∂Δ_{0} × {t_{0}}, τ̃) to (∂Δ_{0} × [t_{0}, t_{1}], τ̃), for any simplex Δ inside one of (∂Δ_{0} × {t_{0}}, τ̃) and (∂Δ_{0} × {t_{1}}, τ̃), we do have

⋂_{x∈Vertex(Δ)} E_{1}(x) ≥ K_{1} − \frac{(n − 1)n}{2},

from our construction (see condition (b) in the proof of Lemma 3.11). But for simplices Δ which are not completely sitting inside one of (∂Δ_{0} × {t_{0}}, τ̃) and (∂Δ_{0} × {t_{1}}, τ̃), we do NOT have

⋂_{x∈Vertex(Δ)} E_{1}(x) ≥ K_{1} − \frac{(n − 1)n}{2},

even if we assume dim(Δ) ≤ n − 1. For the application we have in mind, we need the following strengthened form of Theorem 3.16 (in fact, we will need the version of the following result which allow multiplicities; see Theorem 3.32).

Theorem 3.18. Let (X, σ) be a simplicial complex and Y = X^{(l)} be the lskeleton of X. Suppose that there is a subdivision (Y, τ) of (Y, σ) and a grouping for each vertex of (Y, τ) (and (X, σ)), such that

(a) if Δ is a simplex of (Y, τ), then

# (  ⋂_{y∈Vertex(Δ,τ)} E_{j}(y)) ≥ K_{j} − \frac{l(l + 1)}{2} .  j = 1, 2, ⋯, m;

(b) if Δ is a simplex of (Y, σ) ⊂ (X, σ), and y ∈ Δ is a vertex of (Y, τ), then

⋂_{x∈Vertex(Δ,σ)} E_{j}(x) ⊂ E_{j}(y) ⊂ ⋃_{x∈Vertex(Δ,σ)} E_{j}(x),  j = 1, 2, ⋯, m.

It follows that there is a subdivision (X, τ̃) of (X, σ) and groupings for all the vertices, such that

(1) (X, τ̃)|_{Y} = (Y, τ), and groupings on Vertex(Y, τ) are the same as the old ones.

(2) if Δ is a simplex of (X, σ), and x_{1} ∈ Δ is a newly introduced vertex of (X, τ̃), then

⋂_{x∈Vertex(Δ,σ)} E_{j}(x) ⊂ E_{j}(x_{1}) ⊂ ⋃_{x∈Vertex(Δ,σ)} E_{j}(x),  j = 1, 2, ⋯, m;

(3) for each simplex Δ of (X, τ̃), if Δ is inside the l′-skeleton (X, σ)^{(l′)} (l′ ≥ l) of (X, σ), then

# (  ⋂_{x∈Vertex(Δ,τ̃)} E_{j}(x)) ≥ K_{j} − \frac{l′(l′ + 1)}{2} .  j = 1, 2, ⋯, m.

Proof: If one does not require (1) (i.e., if it is allowed to introduce more vertices into (Y, τ)), then the theorem is Theorem 3.16 (see 3.17 also).

Recall, in the proof of Theorem 3.16, we first constructed a subdivision (X, τ_{1}) and the groupings to make the above (3) hold for E_{1}. Then based on (X, τ_{1}), we constructed a new subdivision (X, τ_{2}) and groupings to make the above (3) hold also for E_{2}, and so on. If we use the same procedure to prove Theorem 3.18, we will encounter a difficulty in the second step. We have no problem for the first step, since we can begin with what we already have on (X^{(l)}, τ) and work on each of the simplexes of dimension larger than l (see Lemma 3.11, the proof of Corollary 3.12 and Remark 3.17). But for the second step, the condition (a) may not hold for l-skeleton (X, τ_{1})^{(l)} of (X, τ_{1}). So we need to start with the simplex of the lowest dimension, which forced us to introduce vertices on (Y, τ) = (X^{(l)}, τ). The following small trick can be used to avoid the difficulty mentioned above. Consider simplex Δ. Suppose that the subdivision (∂Δ, τ̃) and the groupings for those vertices are chosen. Identify Δ with ∂Δ × [0, 1]/∂Δ × {1} as in the proof of Lemma 3.11. Choose a point t_{0} ∈ (0, 1), and write

Δ = ∂Δ × [0, t_{0}] ∪ (∂Δ × [t_{0}, 1]/∂Δ × {1}).

Substitute Δ by Δ^{sub} = ∂Δ × [t_{0}, 1]/∂Δ × {1}. The simplicial structure τ̃_{pre} and the groupings on ∂Δ^{sub} = ∂Δ × {t_{0}} should be endowed the same as τ̃ and the groupings on ∂Δ = ∂Δ×{0}. Then apply Theorem 3.16 to Δ^{sub}. One may introduce new vertices on (∂Δ × {t_{0}}, τ̃_{pre}), but no new vertices are introduced on ∂Δ = ∂Δ × {0}. Finally, for the part ∂Δ × [0, t_{0}], same as in the Step 4 of the proof of Lemma 3.11, we apply Lemma 3.10 to make this part a simplicial complex, in which we do not introduce any new vertices on ∂Δ × {0}.

⊓ ⊔

3.19. For convenience, define

E_{j}(Δ) = ⋂_{x∈Vertex(Δ)} E_{j}(x),  j = 1, 2, ⋯, m

for each simplex Δ of (X, τ̃). Then (3) of 3.18 becomes

#(E_{j}(Δ)) ≥ K_{j} − \frac{l′(l′ + 1)}{2} ,

if Δ is in the l′-skeleton of (X, σ) (l′ ≥ l).

3.20. We need a different version of Theorem 3.18 which allows multiplicity. Let w_{1}, w_{2}, ⋯, w_{k} be a k-tuple of positive integers. Let

E = {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{k}^{∼w_{k}}}

be an index set with multiplicity and λ_{i} ≠ λ_{j} if i ≠ j. (See 1.1.7 (b) for the notation λ^{∼w}.) Let w_{1}+w_{2}+⋯+w_{k} = K. (3.2 is a special case with each w_{i} = 1.) Let K_{1}, K_{2}, ⋯, K_{m} be non negative integers with K_{1} +K_{2} +⋯+K_{m} = K. Suppose that

E_{j} = {λ_{1}^{∼p_{1}^{j}}, λ_{2}^{∼p_{2}^{j}}, ⋯, λ_{k}^{∼p_{k}^{j}}}, j = 1, 2, ⋯, m,

where p_{i}^{j} are nonnegative integers. If {E_{1}, E_{2}, ⋯, E_{m}} satisfies

∑_{i=1}^{k} p_{i}^{j} = K_{j} for each j = 1, 2, ⋯, m, and

∑_{j=1}^{m} p_{i}^{j} = w_{i} for each i = 1, 2, ⋯, k,

then we call {E_{1}, E_{2}, ⋯, E_{m}} a grouping of E of type (K_{1}, K_{2}, ⋯, K_{m}), or just a grouping of E.

3.21. It is convenient to use the notations of union, intersection etc. for the sets with multiplicity. A is called a subset of E if A is of the form

{λ_{1}^{∼t_{1}}, λ_{2}^{∼t_{2}}, ⋯, λ_{k}^{∼t_{k}}}

with 0 ≤ t_{i} ≤ w_{i}, for each 1 ≤ i ≤ k. Note that if all t_{i} = 0, then A is called the empty set. If t_{i} = w_{i}, then A = E. Let B be another subset of E of form {λ_{1}^{∼s_{1}}, λ_{2}^{∼s_{2}}, ⋯, λ_{k}^{∼s_{k}}}. A is a subset of B (denoted by A ⊂ B) if t_{i} ≤ s_{i} for all i. In general, define the union, intersection, and difference of two subsets A and B of E as follows.

A ∪ B = {λ_{1}^{∼max(t_{1},s_{1})}, λ_{2}^{∼max(t_{2},s_{2})}, ⋯, λ_{k}^{∼max(t_{k},s_{k})}}
A ∩ B = {λ_{1}^{∼min(t_{1},s_{1})}, λ_{2}^{∼min(t_{2},s_{2})}, ⋯, λ_{k}^{∼min(t_{k},s_{k})}}
A\B = {λ_{1}^{∼max(0,t_{1}−s_{1})}, λ_{2}^{∼max(0,t_{2}−s_{2})}, ⋯, λ_{k}^{∼max(0,t_{k}−s_{k})}} .

(The definitions of union and intersection can be easily generalized to finitely many subsets of E.) Warning 1: B ∩ (A\B) may be a nonempty set. Warning 2: The assumption that {E_{1}, E_{2}, ⋯ E_{m}} is a grouping of E does NOT imply that ∪_{i=1}^{m}E_{i} = E or that E_{i} ∩ E_{i′} = ∅ for i ≠ i′. (See 3.20.) But A\B = A\(A ∩ B) still holds.

3.22. Let (X, σ) be a simplicial complex. Suppose that each vertex x ∈ X is associated with a grouping {E_{1}, E_{2}, ⋯, E_{m}}, satisfying

#(E_{j}) = ∑_{i=1}^{k} p_{i}^{j} = K_{j}.

(Recall that, for the notation of #, we count multiplicity.) In Theorem 3.18, we introduced subdivisions (X, τ) of (X, σ), and groupings for newly introduced vertices to make #(E_{j}(Δ)) large, for all simplices Δ of (X, τ). In order to prove the decomposition theorem in the next section, we need a stronger result, since the multiplicity of the spectrum of the homomorphism is involved. (See §2. We can not always perturb the map to have distinct spectrum, like the one dimensional case.) Fortunately, this stronger result can be proved in the same way as that for Theorem 3.18, with a few modifications. For any subset F = {λ_{1}^{∼u_{1}}, λ_{2}^{∼u_{2}}, ⋯, λ_{k}^{∼u_{k}}} ⊂ E, define

F^{∘} = {λ_{1}^{∼v_{1}}, λ_{2}^{∼v_{2}}, ⋯, λ_{k}^{∼v_{k}}},

where

v_{i} = \{ \begin{matrix}u_{i} & & if u_{i} = w_{i} \\ & 0 & if u_{i} < w_{i} . \end{matrix}

That is, F^{∘} is the set of all those elements λ_{i}, which are entirely inside F. Evidently,

E_{j}^{∘}(Δ) = ⋂_{x∈Vertex(Δ)}E_{j}^{∘}(x).

Instead of the condition that E_{j}(Δ) is large (see 3.18 and 3.19), we need to make E_{j}^{∘}(Δ) large for any simplex Δ of (X, τ̃). For this purpose, E_{j}^{∘}(x) should be large for each vertex of (X, σ) at the beginning.

3.23. For each set F = {λ_{1}^{∼u_{1}}, λ_{2}^{∼u_{2}}, ⋯, λ_{k}^{∼u_{k}}} ⊂ E, define

F̄ = {λ_{1}^{∼v_{1}}, λ_{2}^{∼v_{2}}, ⋯, λ_{k}^{∼v_{k}}}

where

v_{i} = \{ \begin{matrix}w_{i} & if u_{i} > 0 \\ 0 & if u_{i} = 0 . \end{matrix}

Obviously,

F^{∘} ⊂ F ⊂ F̄.

3.24. Let H_{1}, H_{2}, ⋯, H_{m} (not necessarily disjoint) be finite subsets of E satisfying condition (∗) in 3.7, and E = H_{1} ∪ H_{2} ∪ ⋯ ∪ H_{m}. Suppose that

H_{i}^{∘} = H_{i} = H̄_{i} for each i = 1, 2, ⋯, m.

In what follows, we will require that

E_{j} ⊂ H_{j}, j = 1, 2, ⋯, m

(comparing with the condition (2) in Theorem 3.18).

3.25. For each subset I ⊂ {1, 2, ⋯, m}, define

H_{I} = ⋃_{j∈I} H_{j}.

Let G′_{I} = ∩_{j∈I}H_{j}. Then define

G_{I} = G′_{I}\ ⋃_{J \begin{matrix}⊃ \\ ≠ \end{matrix} I} G′_{J} .

Another way to define G_{I} is by

G_{I} = {λ ∈ E | λ ∈ H_{i} if and only if i ∈ I}.

(Note that G_{I} may be an empty set for some I.) Obviously,

G_{I}^{∘} = G_{I} = Ḡ_{I} ⊂ H_{I}^{∘} = H_{I} = H̄_{I}.

If I ∩ J = ∅, then H_{I} ∩ G_{J} = ∅. Furthermore, for any λ ∈ E, there is a unique set I (defined by I = {i | λ ∈ H_{i}}) such that λ ∈ G_{I}. Hence E is a disjoint union of

{G_{I}, ∅ ≠ I ⊂ {1, 2, ⋯, m}}.

Similarly, we have

H_{I} = ⋃_{J∩I≠∅} G_{J}.

3.26. Under the above partition G_{I} of E, two elements λ, μ ∈ E are in the same part, if and only if the following is true. For any i = 1, 2, ⋯, m, either H_{i} contains both λ and μ, or H_{i} contains none of λ and μ. For any E_{1}, E′_{1} ⊂ H_{1}, if #(E_{1} ∩ G_{I}) = #(E′_{1} ∩ G_{I}) for any I ⊂ {1, 2, ⋯, m}, then from the end of 3.25,

#(E_{1} ∩ H_{I}) = ∑_{J∩I≠∅} #(E_{1} ∩ G_{J}) = ∑_{J∩I≠∅} #(E′_{1} ∩ G_{J}) = #(E′_{1} ∩ H_{I})

for any I ⊂ {1, 2, ⋯, m}. Hence #(H_{I}\E_{1}) = #(H_{I}\E′_{1}). At this circumstance, either both of E_{1} and E′_{1} satisfy (∗∗) in 3.8, or both of them do not satisfy (∗∗) in 3.8. Note that H_{i} = ∪_{I∋i}G_{I}. A grouping {E_{1}, E_{2}, ⋯ E_{m}} satisfies E_{i} ⊂ H_{i} (i = 1, 2, ⋯ m) if and only if for any i ∉ I, E_{i} ∩ G_{I} = ∅. For the rest of the section, let Ω ≥ max(w_{1}, w_{2}, ⋯, w_{m}) be a fixed number, where w_{1}, w_{2}, ⋯, w_{m} are the multiplicities in E. Note that for our application, sometimes, we have to allow Ω to be larger than the maximum multiplicity.

Assumption 3.27. For each grouping {E_{1}(x), E_{2}(x), ⋯, E_{m}(x)}, we always assume that

#(E_{j}^{∘}(x)) ≥ #(E_{j}(x)) − MΩ = K_{j} − MΩ, j = 1, 2, ⋯, m,

where M = 2^{m} − 1. We not only require each initial grouping for (X, σ) to satisfy the above assumption, but also require any new groupings for vertices of (X, τ) to satisfy the assumption. Since M = 2^{m} − 1, there are totally M non-empty subsets I ⊂ {1, 2, ⋯, m}. If the grouping {E_{1}, E_{2}, ⋯, E_{m}} satisfies

#(E_{j}^{∘} ∩ G_{I}) ≥ #(E_{j} ∩ G_{I}) − Ω

for all j = 1, 2, ⋯, m and for all I ⊂ {1, 2, ⋯, m}, then it also satisfies Assumption 3.27.

Lemma 3.28. If {E_{1}, E_{2}, ⋯, E_{m}} is a grouping of E with E_{i} ⊂ H_{i}, then there is a grouping {E′_{1}, E′_{2}, ⋯, E′_{m}} of E satisfying Assumption 3.27, and

E′_{i} ⊂ H_{i}, E′_{i} ⊃ E_{i}^{∘} for all i = 1, 2, ⋯, m.

Proof: The proof is straight forward. Consider E′_{1}, E′_{2}, ⋯, E′_{m} to be m boxes with no element at the beginning, and put each element of E into one of the boxes, following the procedures described below. Step 1. Put all the elements of E_{i}^{∘} into box E′_{i} for each i = 1, 2, ⋯, m. (Thus

E′_{i} ⊃ E_{i}^{∘}.)

Step 2. Fix I ⊂ {1, 2, ⋯, m}. For the set E′_{1}, if there is a λ_{i} ∈ G_{I}\(E′_{1} ∪ E′_{2} ∪ ⋯ ∪ E′_{m}) such that

#(E′_{1} ∩ G_{I}) + w_{i} ≤ #(E_{1} ∩ G_{I}),

where w_{i} is the multiplicity of λ_{i} in E, then put the entire set {λ_{i}^{∼w_{i}}} into E′_{1}. (Note that if 1 ∉ I, then E_{1} ∩ G_{I} = ∅. Hence for I, we need not do anything for E_{1}.) Repeat this procedure until no such i exists. Thus, so far,

#(E′_{1}^{∘} ∩ G_{I}) = #(E′_{1} ∩ G_{I}) ≥ #(E_{1} ∩ G_{I}) − (Ω − 1).

For the same I above, repeat the above construction for the set E′_{2}, then E′_{3}, etc. After this step has been completed for each I, (it is done for each set I separately) we have the following:

#(E′_{j}^{∘} ∩ G_{I}) = #(E′_{j} ∩ G_{I}) ≥ #(E_{j} ∩ G_{I}) − (Ω − 1).

Step 3. Put what left for each G_{I} from the previous steps, arbitrarily into the boxes

E′_{1}, E′_{2}, ⋯, E′_{m}

to make the following condition hold:

#(E′_{j} ∩ G_{I}) = #(E_{j} ∩ G_{I}).

From the end of 3.26, E′_{i} ⊂ H_{i} is a consequence of the above equation. (Note that E_{i} ⊂ H_{i}.) Evidently, {E′_{1}, E′_{2}, ⋯, E′_{m}} is as desired.

⊓ ⊔

3.29. A set E_{1}(⊂ H_{1}) of K_{1} elements is said to satisfy the condition (∗ ∗ ∗), if there is a grouping (E_{1}, E_{2}, ⋯, E_{m}) of E (of type (K_{1}, K_{2}, ⋯ K_{m})), E_{i} ⊂ H_{i}, satisfying Assumption 3.27. Obviously, (∗ ∗ ∗) implies (∗∗). The following corollary is a direct consequence of Lemma 3.28.

Corollary 3.30. For any set E_{1} (⊂ H_{1}) satisfying (∗∗), there is a set E′_{1} (⊂ H_{1}) satisfying (∗ ∗ ∗) such that

E′_{1} ⊃ E_{1}^{∘}.

Proof: Since E_{1} satisfies (∗∗), we can extend E_{1} to a grouping {E_{1}, E_{2}, ⋯, E_{m}} of E such that E_{i} ⊂ H_{i} for each i. By Lemma 3.28, there is a grouping {E′_{1}, E′_{2}, ⋯, E′_{m}} satisfying Assumption 3.27, and E′_{i} ⊂ H_{i} for each i. This is condition (∗ ∗ ∗) for E′_{1}.

⊓ ⊔

Lemma 3.31. Let E_{1} and F_{1} be two sets satisfying condition (∗ ∗ ∗). Suppose that there is a λ ∈ E such that

{λ^{∼w}} ⊂ F_{1}^{∘}\E_{1}^{∘},

where w is the multiplicity of λ in E. Then there are (perhaps repeating)

elements μ_{1}, μ_{2}, ⋯, μ_{t} ∈ E_{1}\F_{1}^{∘}, where t = w − #({λ^{∼w}} ∩ E_{1}), such that

E′_{1} = (E_{1} ∪ {λ^{∼w}})\{μ_{1}, μ_{2}, ⋯, μ_{t}}

satisfies (∗∗) and

\begin{aligned}#((E′_{1}^{∘} ∩E_{1}^{∘}) ∩ G_{I}) & ≥ #(E_{1}^{∘} ∩ G_{I}) − (w + Ω) \\ & ≥ #(E_{1}^{∘} ∩ G_{I}) − 2Ω \end{aligned}

for each I ⊂ {1, 2, ⋯, m}. As a consequence,

#(E′_{1}^{∘} ∩E_{1}^{∘}) ≥ #(E_{1}^{∘}) − 2MΩ.

Proof: Let t_{1} = #({λ^{∼w}} ∩ E_{1}). Then t_{1} < w. Applying Lemma 3.9 t := w − t_{1} times, one can obtain a (possibly repeating) set T′ = {ν_{1}, ν_{2}, ⋯, ν_{t}} ⊂

E_{1}\F_{1} ⊂ E_{1}\F_{1}^{∘}, such that

Ẽ_{1} = (E_{1} ∪ {λ^{∼w}})\T′

satisfies (∗∗). From 3.26, if another set T = {μ_{1}, μ_{2}, ⋯, μ_{t}} ⊂ E_{1}\F_{1}^{∘}, satisfies that

#(T ∩ G_{I}) = #(T′ ∩ G_{I})

for each I ⊂ {1, 2, ⋯, m}, then E′_{1} = (E_{1} ∪ {λ^{∼w}})\T also satisfies (∗∗). T ⊂ E_{1}\F_{1}^{∘} will be constructed to satisfy the following condition. For each I ⊂ {1, 2, ⋯, m},

#(T ∩ G_{I}) = #(T′ ∩ G_{I}),

and (T ∩ (E_{1}\T)) ∩ G_{I} is either empty or {μ_{i}^{∼s}} for a certain μ_{i} ∈ {μ_{1}, μ_{2}, ⋯, μ_{t}}. (Note that T ∩(E_{1}\T) may not be empty, since we are dealing with sets with multiplicities.) To do the above, write

(E_{1}\F_{1}^{∘}) ∩ G_{I} = {λ_{i_{1}}^{∼s_{1}}, λ_{i_{2}}^{∼s_{2}}, ⋯, }.

Then put each of the sets {λ_{i_{1}}^{∼s_{1}}}, {λ_{i_{2}}^{∼s_{2}}}, ⋯, entirely into T one by one until we can not do it without violating the restriction

#(T ∩ G_{I}) ≤ #(T′ ∩ G_{I}).

Then make T to satisfy #(T ∩ G_{I}) = #(T′ ∩ G_{I}) by putting part of {λ_{i_{j}}^{∼s_{j}}} into T if necessary. Since #(T) ≤ w, combining with the above condition for (T ∩ (E_{1}\T)) ∩ G_{I}, after a moment thinking, one can obtain,

#((E_{1}\T)^{∘} ∩ G_{I}) ≥ #(E_{1}^{∘} ∩ G_{I}) − (w + Ω).

(In fact, E_{1}^{∘}\(E_{1}\T)^{∘} ⊂ T^{∘} ∪ (T ∩ (E_{1}\T)), and (T^{∘} ∪ (T ∩ (E_{1}\T))) ∩ G_{I} has at

most w + Ω elements.) Hence

#((E′_{1}^{∘} ∩E_{1}^{∘}) ∩ G_{I}) ≥ #(E_{1}^{∘} ∩ G_{I}) − (w + Ω).

⊓ ⊔

The following is the main result of this section. Together with Lemma 3.28, it will be used in §4.

Theorem 3.32. Let (X, σ) be a simplicial complex, and Y = X^{(l)}, the lskeleton of X. Suppose that (Y, τ) is a subdivision of (Y, σ) and, for each vertex y ∈ Vertex(Y, τ), there is a grouping E_{1}(y), E_{2}(y), ⋯, E_{m}(y) of E (of type (K_{1}, K_{2}, ⋯, K_{m})). Suppose that the groupings satisfy the following three conditions:

(a) For each simplex Δ of (Y, τ), and i = 1, 2, ⋯, m,

#(E_{i}^{∘}(Δ)) ≥ K_{i} − (MΩ + MΩ dim Y · (dim Y + 1)),

where M = 2^{m} − 1.

(b) E_{i}(x) ⊂ H_{i}, i = 1, 2, ⋯, m, for each x ∈ Vertex(Y, τ).

(c) Each grouping, for a vertex of (Y, τ), satisfies Assumption 3.27.

It follows that there exist a subdivision (X, τ̃) of (X, σ) and a grouping for each vertex of (X, τ̃), satisfying the following conditions.

(1) (X, τ̃)|_{Y} = (Y, τ), and each grouping on Vertex(Y, τ) is as same as the old one.

(2) E_{i}(x) ⊂ H_{i}, i = 1, 2, ⋯, m, for each x ∈ Vertex(X, τ̃), and if Δ is a simplex of (X, σ) (before the subdivision), and x ∈ Δ is a newly introduced vertex of (X, τ̃), then

E_{j}^{∘}(x_{1}) ⊃ ⋂_{y∈Vertex(Δ∩Y,τ)}E_{j}^{∘}(y).

(3) For each simplex Δ of (X, τ̃), if Δ is inside the l′-skeleton (X, σ)^{(l′)}, (l′ > l), of (X, σ), then

#(E_{j}^{∘}(Δ)) ≥ K_{j} − (MΩ + MΩl′(l′ + 1)).

(4) Each grouping on Vertex(X, τ̃) satisfies Assumption 3.27.

Proof: (Sketch) The proof is the same as the one of 3.18 (see 3.11 to 3.18), using Lemma 3.31 to replace Lemma 3.9. The arguments in 3.12 – 3.18 are easily adopted in this new setting. We only give the proof for the part corresponding to 3.11 and sketch the differences for other parts. As in 3.11, consider only one simplex X = Δ_{0} with Y = ∂Δ_{0}, and only one set E_{1}(x). Similar to Step 1 of 3.11, choose E_{1}^{model} to satisfy condition (∗ ∗ ∗) and

E_{1}^{∘}^{model} ⊃ ⋂_{x∈Vertex(∂Δ_{0},τ)}E_{1}^{∘}(x).

Replace (a) of 3.11 by

⋂_{x∈Vertex(∂Δ×\{t_{i}\},τ̃)}(E_{1}^{∘}(x)∩E_{1}^{∘}^{model}) \begin{matrix}⊃ \\ ≠ \end{matrix} ⋂_{x∈Vertex(∂Δ×\{t_{i−1}\},τ̃)}(E_{1}^{∘}(x)∩E_{1}^{∘}^{model}) .

Keeping the notations in 3.11, in Step 2, replace G by

G = E_{1}^{∘}(y_{1}, t_{0}) ∩E_{1}^{∘}(y_{2}, t_{0}) ∩ ⋯ ∩E_{1}^{∘}(y_{p}, t_{0}) ∩E_{1}^{∘}^{model}.

If G = E_{1}^{∘}^{model}, then define

E_{1}(y_{i}, t_{1}) = E_{1}^{model}, i = 1, 2, ⋯, p.

Suppose that G ≠ E_{1}^{∘}^{model}. Choose λ ∈ E_{1}^{∘}^{model}\G. Let w be the multiplicity of λ. Then

{λ^{∼w}} ⊂ E_{1}^{∘}^{model}\G.

For each (y_{i}, t_{0}), if λ ∈ E_{1}^{∘}(y_{i}, t_{0}), then define E_{1}(y_{i}, t_{1}) = E_{1}(y_{i}, t_{0}) (as in Step 2 of 3.11). If λ ∉ E_{1}^{∘}(y_{i}, t_{0}), apply Lemma 3.31 to obtain E′_{1} satisfying(∗∗),

E′_{1}^{∘} ⊃ G ∪ {λ^{∼w}} and

(A)  #(E′_{1}^{∘} ∩E_{1}^{∘}) ≥ #(E_{1}^{∘}) − 2MΩ.

Then we can apply Corollary 3.29 to find E′′_{1} satisfying (∗ ∗ ∗) and E′′_{1}^{∘} ⊃ E′_{1}^{∘}. Define

E_{1}(y_{i}, t_{1}) = E′′_{1}.

Then

E_{1}^{∘}(y_{i}, t_{1}) ⊃ E′_{1}^{∘} ⊃ G ∪ {λ^{∼w}}.

The arguments in Step 3 and Step 4 of 3.11 can also be employed here. (Of course, at many places (not all places), one needs to replace E_{i} by E_{i}^{∘}.) The estimation (e) in Step 4 of 3.11 will be changed to

#(E_{1}^{∘}(u_{0}, t_{0})∩E_{1}^{∘}(u_{1}, t_{0})∩⋯∩E_{1}^{∘}(u_{i}, t_{0})∩E_{1}^{∘}(u_{0}, t_{1})∩E_{1}^{∘}(u_{1}, t_{1})∩⋯∩E_{1}^{∘}(u_{i}, t_{1}))

≥ K_{1} − [MΩ + MΩ dim Y · (dim Y + 1)] − 2MΩ(i + 1)

= K_{1} − [MΩ + MΩ · (n − 1) · n] − 2MΩ · n
= K_{1} − [MΩ + MΩ · n · (n + 1)].

(Here we used the above estimation (A) which is from Lemma 3.31.) Since E_{1}(y_{i}, t_{1}) satisfies (∗ ∗ ∗), all the other parts (e.g., induction arguments) in 3.11—3.18 can go through easily. In the part corresponding to the proof of Corollary 3.14, the definition of D_{i} should be changed to

D_{i} = ⋂_{x∈Vertex(Δ)}E_{i}^{∘}(x).

⊓ ⊔

We remark that in §4, we will only use the theorem of the case that X = Δ, a single simplex with Y = ∂Δ.

Remark 3.33. The condition E_{i}(x) ⊂ H_{i} in (2) can be strengthened as

E_{i}(x) ⊂ ⋃_{y∈Vertex(Δ∩Y,τ)} Ē_{i}(y),

where Δ is a simplex of (X, σ) (before the subdivision) such that x ∈ Δ is a newly introduced vertex. (The notation is from 3.23.)

4 Decomposition Theorems

In this section, we will prove the decomposition theorems which are needed for the proof of our main Reduction Theorem and the main results in [EGL]. The following Theorem 4.1 is one version of the Decomposition Theorem. After Theorem 4.1 has been proved, we will use [Li 2] to verify that the condition of Theorem 4.1 holds for connecting homomorphisms ϕ_{n,m}(for each fixed n, m should be large enough), with the maps a_{1}, a_{2}, ⋯ a_{L}(see below) factoring through interval [0, 1] or the single point space {pt}. In such a way, we can prove our main decomposition theorems (Theorem 4.35 and Theorem 4.37).

Theorem 4.1. Let X be a connected finite simplicial complex, and F ⊂ C(X) be a finite set which generates C(X). For any ε > 0, there is an η > 0 such that the following statement is true. Suppose that a unital homomorphism ϕ : C(X) → P M_{K′}(C(Y))P (rank(P) = K) (where Y is a finite simplicial complex) satisfies the following condition: There are L continuous maps

a_{1}, a_{2}, ⋯, a_{L} : Y ⟶ X

such that for each y ∈ Y, SPϕ_{y} and Θ(y) can be paired within η, where

Θ(y) = {a_{1}(y)^{∼T_{1}}, a_{2}(y)^{∼T_{2}}, ⋯, a_{L}(y)^{∼T_{L}}}

and T_{1}, T_{2}, ⋯, T_{L} are positive integers with

T_{1} + T_{2} + ⋯ + T_{L} = K = rank(P).

(See 1.1.7(b) for notation x^{∼T_{i}}.) Let T = 2^{L}(dim X + dim Y)^{3}. It follows that there are L mutually orthogonal projections p_{1}, p_{2}, ⋯, p_{L} ∈ P M_{K′}(C(Y))P such that

(i) ∥ϕ(f)(y)−p_{0}(y)ϕ(f)(y)p_{0}(y)⊕∑_{i=1}^{L} f(a_{i}(y))p_{i}(y)∥ < ε, for any f ∈ F and y ∈ Y, where p_{0} = P − ∑_{i=1}^{L} p_{i}; (ii) ∥p_{0}(y)ϕ(f)(y) − ϕ(f)(y)p_{0}(y)∥ < ε for any f ∈ F and y ∈ Y; (iii) rank(p_{i}) ≥ T_{i} − T for 1 ≤ i ≤ L, and hence rank(p_{0}) ≤ LT.

4.2. In the above theorem, some of the p_{i} may be zero projections if T_{i} ≤ T. But when the theorem is applied later in this article, the positive integers T_{i} are always very large compared with T = 2^{L}(dim X + dim Y)^{3}. The proof of this theorem will be divided into several small steps. The results in §2 and §3 will be used. In fact, §2 and §3 will only be used in the proof of Theorem 4.1, no other place in this paper or [EGL]. (The results in §2 have some other applications.)

4.3. The theorem is trivial if X = {pt}, applying Theorem 1.2 of [Hu,Chapter 8]. Without loss of generality, we assume that X ≠ {pt}. By the results in §2, we can assume that ϕ has maximum spectral multiplicity at most Ω := dim X + dim Y.

4.4. For any ε > 0, there is an η > 0 such that for any x_{1}, x_{2} ∈ X, if dist(x_{1}, x_{2}) < 2η, then

|f(x_{1}) − f(x_{2})| < \frac{ε}{3} for all f ∈ F.

We will prove that this η is as desired.

4.5. Recall, from 1.2.5, for any positive integer n, P^{n}X is the symmetric product of n-copies of X. Also, any element Λ ∈ P^{K}X can be considered as a set with multiplicity. So SPϕ_{y} ∈ P^{K}X. Suppose that Λ_{1} ∈ P^{k_{1}}X, Λ_{2} ∈ P^{k_{2}}X, ⋯, Λ_{t} ∈ P^{k_{t}}X. Write

Λ_{1} = {λ_{1}, λ_{2}, ⋯, λ_{k_{1}}}
Λ_{2} = {λ_{k_{1}+1}, λ_{k_{1}+2}, ⋯, λ_{k_{1}+k_{2}}}
⋮
Λ_{t} = {λ_{k_{1}+···+k_{t−1}+1}, ⋯, λ_{k_{1}+···+k_{t}}}

as sets with multiplicity. By abusing the notation, we use {Λ_{1}, Λ_{2}, ⋯, Λ_{t}} to denote

{λ_{1}, λ_{2}, ⋯, λ_{k_{1}}, λ_{k_{1}+1}, ⋯, λ_{k_{1}+k_{2}}, · · · · · · , λ_{k_{1}+k_{2}+···+k_{t}}} ,

which defines an element in P^{k_{1}+k_{2}+···+k_{t}}X. (Note that {Λ_{1}, Λ_{2}, ⋯, Λ_{t}} = Λ_{1} ∪ Λ_{2} ∪ ⋯ ∪ Λ_{t}, if {Λ_{j}} are mutually disjoint. See 3.21 for the definition of unions of sets with multiplicity.)

4.6. For any fixed point y ∈ Y, write

SPϕ_{y} = {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{k}^{∼w_{k}}}

with λ_{i} ≠ λ_{j} if i ≠ j. Note that w_{1} + w_{2} + ⋯ + w_{k} = K. We denote the above k by t_{y} to indicate that this integer depends on y. Define

(a)  θ(y) = \frac{1}{4} min_{1≤i<j≤t_{y}} dist(λ_{i}, λ_{j}).

Then θ(y) > 0 for any y ∈ Y. (Of course, θ(y), in general, does not depend on y continuously.) For each i = 1, 2, ⋯, t_{y}, there is an open set U(y, i) ∋ λ_{i} such that

(b)  diameter(U(y, i)) ≤ min (\frac{η}{2(dim Y + 1)}, θ(y)) .

Then, obviously,

(c)  dist(U(y, i), U(y, j)) ≥ 2θ(y)  if  i ≠ j.

Applying Lemma 1.2.10, there is an (connected) open neighborhood O(y) of y such that

SPϕ_{y′} ⊂ U(y, 1) ⋃ U(y, 2) ⋃ ⋯ ⋃ U(y, t_{y})

for all y′ ∈ O(y). Define the continuous maps

Λ_{1} : O(y) ⟶ P^{w_{1}}U(y, 1) (⊂ P^{w_{1}}X)
Λ_{2} : O(y) ⟶ P^{w_{2}}U(y, 2) (⊂ P^{w_{2}}X)
⋮
Λ_{t_{y}} : O(y) ⟶ P^{w_{ty}} U(y, 1) (⊂ P^{w_{ty}} X)

by Λ_{i}(y′) = SPϕ_{y′} ∩ U(y, i). Then

SPϕ_{y′} = {Λ_{1}(y′), Λ_{2}(y′), ⋯, Λ_{t_{y}}(y′)}

for each y′ ∈ O(y). Later on, we will use the disjoint open cover

U(y, 1) ⋃ U(y, 2) ⋃ ⋯ ⋃ U(y, t_{y}) ⊃ SPϕ_{y′}

of SPϕ_{y′} to decompose SPϕ_{y′} into a disjoint union of SPϕ_{y′} ∩ U(y, t) and to identify the elements in each set SPϕ_{y′} ∩ U(y, t) as a single element with multiplicity w_{t}. We further require that O(y) is so small that

(c)  diameter(a_{i}(O(y))) ≤ \frac{η}{2(dim Y + 1)},

where a_{i} : Y → X is any one of the continuous maps a_{1}, a_{2}, ⋯, a_{L}, appeared in Theorem 4.1.

4.7. Considering the open cover {O(y)}_{y∈Y} of Y, where the open sets O(y) are from 4.6, there exists a finite sub-cover

O = {O_{1}, O_{2}, ⋯, O_{•}} ⊂ {O(y)}_{y∈Y} ,

of Y. Without loss of generality, we assume that the simplicial complex structure (Y, σ) of Y satisfies the following condition, because we can always refine it if necessary. For each simplex Δ of (Y, σ), the closure of

Star(Δ) := ⋃_{Δ′∩Δ≠∅} interior(Δ′)

can be covered by an open set O_{i} ∈ O, where

interior(Δ′) = Δ′\∂Δ′.

(Note that Star(Δ) is an open set, see 1.4.2)

4.8. For each y ∈ Y, in order to construct p_{1}(y), p_{2}(y), ⋯, p_{L}(y), as in Theorem 4.1, we need to split SPϕ_{y} into L sets E_{1}(y), E_{2}(y), ⋯, E_{L}(y) such that each set E_{i}(y) is contained in an open ball of a_{i}(y) with small radius (smaller than 2η) and that #(E_{i}(y)) = T_{i}, 1 ≤ i ≤ L, where a_{i} and T_{i} are maps and positive integers appeared in Theorem 4.1. Since E_{1}(y), E_{2}(y), ⋯, E_{L}(y) may have non-empty intersection (because of the multiplicity of the spectrum), we need to introduce certain subsets of them, which are E_{1}^{∘} ⊂ E_{1}(y), E_{2}^{∘} ⊂ E_{2}(y), ⋯, E_{L}^{∘} ⊂ E_{L}(y), in the notations of 3.22. This will become precise when the index set with multiplicity is introduced later. The projections p_{1}(y), p_{2}(y), ⋯, p_{L}(y), to be constructed, will be certain sub-projections of the spectral projections corresponding to E_{1}^{∘},E_{2}^{∘}, ⋯,E_{L}^{∘}, respectively. (See Definition 1.2.8 for the spectral projection.) Following §3, a split of SPϕ_{y} into L sets E_{1}(y), E_{2}(y), ⋯, E_{L}(y) will be called a grouping of SPϕ_{y}. The word “grouping” is reserved only for this purpose. Recall, from 4.6, SPϕ_{y′} can be written as a disjoint union

SPϕ_{y′} = (SPϕ_{y′} ∩ U(y, 1)) ⋃(SPϕ_{y′} ∩ U(y, 2)) ⋃ ⋯ ⋃(SPϕ_{y′} ∩ U(y, t_{y})).

And the elements in each set SPϕ_{y′} ∩U(y, t) can be identified as a single element with multiplicity. This will serve as the index set for the groupings. To avoid confusion, the above decomposition is NOT called a “grouping” of SPϕ_{y′}. It is called a decomposition instead. In the next few paragraphs, we apply §3 to construct a subdivision (Y, τ) of (Y, σ) and useful groupings for all vertices y ∈ Vertex(Y, τ).

4.9. Let Δ be a simplex of (Y, σ) and

O(y_{1}), O(y_{2}), ⋯, O(y_{i}),

the list of all open sets in O, each of which covers Δ. Suppose that

θ(y_{1}) ≤ θ(y_{2}) ≤ ⋯ ≤ θ(y_{i}).

From 4.6, for any y ∈ Δ ⊂ ∩_{k=1}^{i}O(y_{k}), and j ∈ {1, 2, ⋯, i},

SPϕ_{y} ⊂ U(y_{j}, 1) ⋃ U(y_{j}, 2) ⋃ ⋯ ⋃ U(y_{j}, t_{y_{j}}).

Claim: If j < j′ ∈ {1, 2, ⋯, i}, then each open set U(y_{j}, t) (t = 1, 2 ⋯, t_{y_{j}})

intersects with at most one of {U(y_{j′}, s)}_{s=1}^{t_{y_{j′}}}. Proof of the Claim: Suppose that the claim is not true, that is, for some t ∈ {1, 2, ⋯, t_{y_{j}}}, there are two different s_{1}, s_{2} ∈ {1, 2 ⋯, t_{y_{j′}} } such that

U(y_{j}, t) ∩ U(y_{j′}, s_{1}) ≠ ∅ and U(y_{j}, t) ∩ U(y_{j′}, s_{2}) ≠ ∅.

Together with the fact that diameter(U(y_{j}, t)) ≤ θ(y_{j}) (see (b) in 4.6), it yields

dist(U(y_{j′}, s_{1}), U(y_{j′}, s_{2})) ≤ θ(y_{j}).

This contradicts with (c) in 4.6 which gives

dist(U(y_{j′}, s_{1}), U(y_{j′}, s_{2})) ≥ 2θ(y_{j′}) > θ(y_{j}).

(Recall θ(y_{j}) ≤ θ(y_{j′}).) This proves the claim. Still suppose that j < j′. From the claim, we have the following. For each y ∈ Δ, if two different elements of SPϕ_{y} are identified as a single element in the decomposition

U(y_{j}, 1) ⋃ U(y_{j}, 2) ⋃ ⋯ ⋃ U(y_{j}, t_{y_{j}})

(i.e., if these two elements are in the same open set U(y_{j}, t) for some t ∈ {1, 2, ⋯, t_{y_{j}}}), then these two elements are also identified as a single element in the decomposition

U(y_{j′}, 1) ⋃ U(y_{j′}, 2) ⋃ ⋯ ⋃ U(y_{j′}, t_{y_{j′}})

(i.e., these two elements are also in the same open set U(y_{j′}, s) for some s ∈ {1, 2, ⋯, t_{y_{j′}} }). Therefore, the decompositions of SPϕ_{y} corresponding to y_{1} and y_{i} are the finest and coarsest decompositions, respectively, among all the above decompositions (corresponding to y_{1}, y_{2}, ⋯, y_{i}). The coarsest decomposition will be used to decompose SPϕ_{y} into several sets. The elements in each of the sets will be identified as a single element with multiplicity. Denote θ(y_{i}) by θ(Δ). (Recall that O(y_{1}), O(y_{2}), ⋯, O(y_{i}) is the list of all open sets in O, each of which covers Δ. Therefore, θ(y_{i}) — the maximum of all {θ(y_{j})}_{j=1}^{i} — depends only on Δ.) Introduce the following notations.

Λ(Δ, 1)(y) = U(y_{i}, 1) ∩ SPϕ_{y},
Λ(Δ, 2)(y) = U(y_{i}, 2) ∩ SPϕ_{y},
⋮
Λ(Δ, t_{Δ})(y) = U(y_{i}, t_{y_{i}}) ∩ SPϕ_{y},

where t_{Δ} = t_{y_{i}}. Recall (see 4.6) that SPϕ_{y_{i}} is written as

SPϕ_{y_{i}} = {λ_{1}^{∼w_{1}}, λ_{2}^{∼w_{2}}, ⋯, λ_{k}^{∼w_{k}}},

where k = t_{Δ} = t_{y_{i}}. Since y ∈ Δ ⊂ O(y_{i}),

#(Λ(Δ, t)(y)) = w_{t}, 1 ≤ t ≤ t_{Δ},

counting multiplicity. Define set

Λ(Δ) = {Λ(Δ, 1)^{∼w_{1}}, Λ(Δ, 2)^{∼w_{2}}, ⋯, Λ(Δ, k)^{∼w_{k}}},

where k = t_{Δ}. That is, identify all the elements of SPϕ_{y} in Λ(Δ, t)(y) as a single element (denoted by Λ(Δ, t)) with the multiplicity. As above, we will use Λ(Δ, t)(y) for two purposes. It is a subset of SPϕ_{y}, or it is a single element in Λ(Δ) which repeats w_{t} times. Strictly speaking, w_{t} (t = 1, 2, ⋯, t_{Δ}) should be written as w_{t}(Δ), and the set Λ(Δ) should be written as {Λ(Δ, 1)^{∼w_{1}(Δ)}, Λ(Δ, 2)^{∼w_{2}(Δ)}, ⋯, Λ(Δ, k)^{∼w_{k}(Δ)}}. When there is a danger of confusion, we will use w_{t}(Δ) instead of w_{t}.

4.10. Let Y′ ⊂ Y be a path connected subspace. Usually we will let Y′ be either an open or a closed subset. Suppose that there are positive integers u_{1}, u_{2}, ⋯, u_{t} and continuous maps

A(Y′, i) : Y′ → P^{u_{i}}X, i = 1, 2, ⋯, t,

such that {SPϕ_{y}}_{y∈Y′} can be decomposed as

SPϕ_{y} = {A(Y′, 1)(y), A(Y′, 2)(y), ⋯, A(Y′, t)(y)}

for all y ∈ Y′. We say that the above decomposition of {SPϕ_{y}}_{y∈Y′} satisfies the condition (S) (S stands for separation) if (S): there are mutually disjoint open sets U_{1}, U_{2}, ⋯, U_{t} ⊂ X satisfying

A(Y′, i)(y) ⊂ U_{i}, ∀y ∈ Y′, i = 1, 2, ⋯, t.

Define

A(Y′) = {A(Y′, 1)^{∼u_{1}}, A(Y′, 2)^{∼u_{2}}, ⋯, A(Y′, t)^{∼u_{t}}},

where u_{i} = #(A(Y′, i)), counting multiplicity. Again, A(Y′, s)(y) is used for two purposes. It is regarded as a subset of SPϕ_{y} or as a single element of A(Y′) with multiplicity u_{s}. In fact, if U_{1}, U_{2}, ⋯ U_{t} are open sets, with mutually disjoint closure, such that SPϕ_{y} ⊂ U_{1} ∪ U_{2} ∪ ⋯ ∪ U_{t}, ∀y ∈ Y′, then #(SPϕ_{y} ∩ U_{i}), y ∈ Y′ are constants, denoted by u_{i}, (note that Y′ is path connected). Furthermore, the maps

A(Y′, i) : Y′ → P^{u_{i}}X, i = 1, 2, ⋯ t,

defined by A(Y′, i)(y) = SPϕ_{y} ∩ U_{i}, are continuous, and they determine a decomposition of {SPϕ_{y}}_{y∈Y′} as

SPϕ_{y} = {A(Y′, 1)(y), A(Y′, 2)(y), ⋯, A(Y′, t)(y)}

satisfying the condition (S). (See Lemmas 1.2.9 and 1.2.10.) For any y ∈ Y′, a grouping E_{1}, E_{2}, ⋯, E_{L} of SPϕ_{y} induces a UNIQUE grouping

E_{1}^{A(Y′)}, E_{2}^{A(Y′)}, ⋯ EA(Y′)_{L} of A(Y′), defined by

E_{i}^{A(Y′)} = {A(Y′, 1)^{∼v_{1}}, A(Y′, 2)^{∼v_{2}}, ⋯, A(Y′, t)^{∼v_{t}}}, i = 1, 2, ⋯, L,

where v_{j} = #(E_{i} ∩ A(Y′, j)(y)), counting multiplicity. (Here the intersection of sets is defined as for the sets with multiplicity, as in 3.21.) On the other hand, let E_{1}, E_{2}, ⋯, E_{L} be a grouping of A(Y′). Define a grouping of the set SPϕ_{y}, for any y ∈ Y′, in the following way. For any j = 1, 2, ⋯, L, if the part E_{j} (for the grouping of A(Y′)) contains exactly w elements of {A(Y′, s)^{∼u_{s}}} (w ≤ u_{s}), then the part E_{j} (for the grouping of the set of SPϕ_{y}) contains exactly w elements (counting multiplicity) which are contained in A(Y′, s)(y). Since these w elements are to be chosen, the induced grouping is not unique. But we will always fix one of them for use. Let E_{1}, E_{2}, ⋯, E_{L} be a grouping of

A(Y′) = {A(Y′, 1)^{∼u_{1}}, A(Y′, 2)^{∼u_{2}}, ⋯, A(Y′, t)^{∼u_{t}}}.

Define E_{1}^{∘},E_{2}^{∘}, ⋯,E_{L}^{∘} as in 3.22.

Although the subsets of SPϕ_{y} corresponding to E_{i} are not unique, the subsets of SPϕ_{y} corresponding to E_{i}^{∘} are unique. We denote them by E_{i}^{∘}|_{y}. Also, #(E_{i}^{∘}) = #(E_{i}^{∘}|_{y}) counting multiplicity. Note that we use E_{i}^{∘}|_{y} instead of E_{i}^{∘}(y) for the following reason (also see the next paragraph). We reserve the notation {E_{i}(y)}i=1^{L} for the grouping of SPϕ_{y} which is associated to a vertex y in a certain simplicial complex (Y, τ). (τ is a subdivision of σ.) Suppose that y ∈ Y′. Let E_{1}(y), E_{2}(y), ⋯ E_{L}(y) be a grouping of SPϕ_{y}. Then

it induces a grouping E_{1}^{A(Y′)}(y), E_{2}^{A(Y′)}(y), ⋯ EA(Y′)_{L}(y) of A(Y′) as above. The sets ^{∘} E_{i}^{A(Y′)}(y) are well defined as subsets of A(Y′). (Warning: ^{∘} E_{i}^{A(Y′)}(y) are not subsets of SPϕ_{y}.) Also, from the last paragraph, the sets E_{i}^{A(Y)}^{∘}(y)|_{z} are well defined as subsets of SPϕ_{z} for any z ∈ Y′ (may be different from y). Furthermore, #(^{∘} E_{i}^{A(Y′)}(y)) = #(^{∘} E_{i}^{A(Y′)}(y)|_{z}). In the next few paragraphs, each grouping of SPϕ_{y} can be referred as a grouping of A(Y′) for different space Y′ and different decomposition A(Y′) provided that y ∈ Y′, or vise versa.

4.11. Note that the collection of sets {Λ(Δ, i)}_{i=1}^{t_{Δ}} (in 4.9) could be regarded as a decomposition of SPϕ_{y}, y ∈ Δ (see 4.6 also). And this decomposition satisfies condition (S) for Δ in place of Y′; therefore 4.10 can be applied to Λ(Δ) as A(Y′). As mentioned in 4.8, we will introduce the groupings of SPϕ_{y} for all vertices of a certain subdivision of (Y, σ). As in section 3, for a simplex Δ of (Y, σ), once we have the subdivision (∂Δ, τ) of (∂Δ, σ) and groupings for all vertices in Vertex(∂Δ, τ), then we can define the subdivision (Δ, τ) of (Δ, σ) and introduce the groupings for all newly introduced vertices. One may notice that, in section 3, for different vertices, the index sets involved are the same. But in the setting here, the index sets SPϕ_{y} are different for different vertices y. So some special care should be taken. Suppose that (Δ, σ) is a simplicial complex consisting of a single simplex Δ and all its faces. Suppose that there is a subdivision (∂Δ, τ) of (∂Δ, σ) and the groupings E_{1}(y), E_{2}(y), ⋯, E_{L}(y) of SPϕ_{y} for all vertices y ∈ Vertex(∂Δ, τ) (see notation in 3.1–3.3). Attention: When we introduce a grouping E_{1}(z), E_{2}(z), ⋯, E_{L}(z) of SPϕ_{z} for any newly introduced vertex z ∈ interior(Δ) = Δ\∂Δ, the following procedure will always be used. First, as in 4.10, we can regard the groupings E_{1}(y), E_{2}(y), ⋯, E_{L}(y) of SPϕ_{y} as groupings E_{1}^{Λ(Δ)}(y), E_{2}^{Λ(Δ)}(y), ⋯, EΛ(Δ)_{L}(y) of Λ(Δ) for all vertices y ∈ Vertex(∂Δ, τ). (Then the set E_{i}^{Λ(Δ)}(y)∩E_{i}^{Λ(Δ)}(y′)∩⋯, as a subset of Λ(Δ), makes sense, for vertices y, y′, ⋯ ∈ Vertex(∂Δ, τ). Also {E_{i}^{Λ(Δ)}^{∘}(y)}i=1^{L} are subsets of Λ(Δ).) Then we use these groupings of the same index set, Λ(Δ), applying the results from section 3 (see 3.32), to introduce subdivision (Δ, τ) of (Δ, σ) and groupings of Λ(Δ) for all newly introduced vertices z ∈ Δ\∂Δ. Finally, theses groupings of Λ(Δ) will induce the groupings E_{1}(z), E_{2}(z), ⋯, E_{L}(z) of SPϕ_{z} as in 4.10 (not unique, but we fix one of them for our use). Furthermore, as in 4.10, E_{1}^{Λ(Δ)}^{∘}(z), E_{2}^{Λ(Δ)}^{∘}(z), ⋯, EΛ(Δ)_{L}^{∘}(z) are well defined subsets of Λ(Δ) and E_{1}^{Λ(Δ)}^{∘}(z)|_{z′}, E_{2}^{Λ(Δ)}^{∘}(z)|_{z′}, ⋯, EΛ(Δ)_{L}^{∘}(z)|_{z′} are well defined subsets of SPϕ_{z′} for any z′ ∈ Δ (not necessarily a vertex).

4.12. Let Δ′ be a face of Δ. Then for y ∈ Δ′ ⊂ Δ, both Λ(Δ′) and Λ(Δ) can be viewed as decompositions of SPϕ_{y}. Recall, in 4.9, the decomposition corresponding to Λ(Δ) is the coarsest decomposition among those corresponding to O(y_{j}) such that O(y_{j}) ⊃ Δ and that O(y_{j}) ∈ O. Since Δ′ ⊂ Δ, any open set in O which covers Δ will also cover Δ′. Therefore, the decomposition of SPϕ_{y} corresponding to Λ(Δ′) is coarser than that corresponding to Λ(Δ). That is, each set Λ(Δ′, s)(y) is a finite union of certain sets,

Λ(Δ, t_{1})(y) ⋃ Λ(Δ, t_{2})(y) ⋃ ⋯,

as subsets of SPϕ_{y}. (Notice that if w′_{s} = w_{s}(Δ′) is the multiplicity appeared in Λ(Δ′) for Λ(Δ′, s) and w_{t} = w_{t}(Δ) is the multiplicity appeared in Λ(Δ) for Λ(Δ, t), then w′_{s} = w_{t_{1}} + w_{t_{2}} + ⋯, a finite sum.) It follows that if y ∈ Δ′ ⊂ Δ and E_{1}(y), E_{2}(y), ⋯, E_{L}(y) is a grouping of SPϕ_{y}, then

∘ E_{j}^{Λ(Δ′)}(y)|_{y′} ⊂ E_{j}^{Λ(Δ)}^{∘}(y)|_{y′},

regarded as subsets of SPϕ_{y′} for any y′ ∈ Δ′. Now we are ready to construct the subdivision (Y, τ) of (Y, σ), and the grouping for each vertex of (Y, σ) and each vertex of the complex (Y, τ) (after subdivision). Since the notations y_{1}, y_{2}, ⋯, y_{•} have been already used for the open cover O = {O(y_{1}), O(y_{1}), ⋯, O(y_{•})}, we use z_{1}, z_{2}, ⋯ to denote the points in Y, especially the vertices of certain simplicial structure.

4.13. Let Ω = dim X + dim Y, M = 2^{L} − 1, where L is the number of the continuous maps {a_{i}} appeared in the statement of 4.1. Note that all the multiplicities w_{t} appearing in any of Λ(Δ) do not exceed Ω, by 4.3 and the construction of Λ(Δ) (see 4.6 and 4.9). For each vertex z ∈ Vertex(Y, σ), by the condition of Theorem 4.1, SPϕ_{z} and

Θ(z) = {a_{1}(z)^{∼T_{1}}, a_{2}(z)^{∼T_{2}}, ⋯, a_{L}(z)^{∼T_{L}}}

can be paired within η. Therefore, we can define a grouping E_{pre,1}(z), E_{pre,2}(z), ⋯, E_{pre,L}(z) of SPϕ_{z}, with T_{1}, T_{2}, ⋯, T_{L} elements, respectively, counting multiplicity, such that

(1)  dist(λ, a_{i}(z)) < η

if λ ∈ E_{pre,i}(z), where η is as in 4.4. (We denote them by E_{pre,i} because this grouping will be modified later.) We can regard such a grouping of SPϕ_{z} as a grouping of Λ(Δ), where Δ ∋ z is a simplex. First we regard it as the grouping of Λ({z}), where {z} is the 0-dimensional simplex of (Y, σ) corresponding to vertex z. By Lemma 3.28, we can modify the grouping to satisfy the Assumption 3.27. Then this modified grouping of Λ({z}) could induce a grouping on SPϕ_{z}, for which the condition (1) above may not hold. But if we carefully choose the sets H_{i} in Lemma 3.28, we could still guarantee that any elements λ ∈ E_{i} are close to a_{i}(z) (see (2) below). In this subsection, we will also introduce the sets H_{i}(Δ) to serve as the sets H_{i} of Lemma 3.28 and Theorem 3.32, when we construct groupings on Δ from the groupings on ∂Δ, by applying Theorem 3.32.

For each vertex z_{0} ∈ Vertex(Y, σ), the notation {z_{0}} is used to denote the corresponding zero dimensional simplex of (Y, σ). The above grouping induces a grouping {E_{pre,i}^{Λ({z_{0}})}} of

Λ({z_{0}}) = {Λ({z_{0}}, 1)^{∼w_{1}}, Λ({z_{0}}, 2)^{∼w_{2}}, ⋯, Λ({z_{0}}, t_{{z_{0}}})^{∼w_{t_{{z_{0}}}}} }.

Define subsets H_{1}({z_{0}}), H_{2}({z_{0}}), ⋯, H_{L}({z_{0}}) of Λ({z_{0}}) as follows. For any i = 1, 2, ⋯, L, H_{i}({z_{0}}) is the collection of all Λ({z_{0}}, t)^{∼w_{t}} (⊂ Λ({z_{0}})) satisfying

Λ(\{z_{0}\}, t)(z_{0}) ⊂ \{x : dist(x, a_{i}(z_{0})) < η + \frac{1}{(dim Y + 1)} · η\} .

Note that each set Λ({z_{0}}, t)(z_{0}) (as a subset of certain U(y_{j}, t), from 4.9) has diameter at most \frac{η}{2(dim Y +1)} < \frac{η}{(dim Y +1)} (see (b) in 4.6). Combining this fact with (1) above, we know that if λ_{1} ∈ E_{pre,i}(z_{0}) and λ_{1} ∈ Λ({z_{0}}, t)(z_{0}), then

(2)  dist(λ, a_{i}(z_{0})) < η + \frac{η}{(dim Y + 1)}

for any λ ∈ Λ({z_{0}}, t)(z_{0}). That means E_{pre,i}^{Λ({z_{0}})} ⊂ H_{i}({z_{0}}). By Lemma 3.28, the above grouping can be modified to another grouping E_{i}^{Λ({z_{0}})} of Λ({z_{0}}) satisfying

#(E_{i}^{∘}^{Λ({z_{0}})}) ≥ T_{i} − MΩ.

(This is Assumption 3.27.) And E_{i}^{Λ({z_{0}})} ⊂ H_{i}({z_{0}}) still holds, regarded as a grouping of Λ({z_{0}}). The above grouping of Λ({z_{0}}) could induce a grouping E_{1}(z_{0}), E_{2}(z_{0}), ⋯, E_{L}(z_{0}) of SPϕ_{z_{0}} (see 4.10 and 4.11). This grouping will be used as the grouping for vertex z_{0}. Even though (1) may not hold for λ in the new E_{i}(z_{0}), (2) holds for any λ in the new E_{i}(z_{0}), from the definition of H_{i}({z_{0}}), and E_{i}^{Λ({z_{0}})} ⊂ H_{i}({z_{0}}). For each simplex Δ of (Y, σ), let us also define the subsets H_{1}(Δ), H_{2}(Δ), ⋯, H_{L}(Δ) of Λ(Δ) as follows. For each j = 1, 2, ⋯, L, H_{j}(Δ) is the collection of all such Λ(Δ, t)^{∼w_{t}} (⊂ Λ(Δ)) that Λ(Δ, t)(z), as a subset of SPϕ_{z}, satisfies

Λ(Δ, t)(z) ⊂ \{x : dist(x, a_{i}(z)) < η + \frac{dim(Δ) + 1}{(dim Y + 1)} · η\}

for any z ∈ Δ. These sets will serve as the sets H_{1}, H_{2}, ⋯, H_{L} when we apply Theorem 3.32. The following fact follows directly from the definition of Λ(Δ, t) and H_{i}(Δ), which will be used in 4.14: Suppose that z ∈ Δ. A grouping E_{1}, E_{2}, ⋯ E_{L} of SPϕ_{z}, regarded as a grouping of Λ(Δ), satisfies E_{i} ⊂ H_{i}(Δ) if and only if for any λ ∈ E_{i} (as a subset of SPϕ_{z}), for the index t satisfying λ ∈ Λ(Δ, t)(z) (such t exists; see 4.9), we have {Λ(Δ, t)^{∼w_{t}(Δ)}} ⊂ H_{i}(Δ).

4.14. Beginning with the simplicial structure (Y, σ) and the above groupings for all 0-dimensional simplex (i.e., vertex) of (Y, σ), we will construct a subdivision (Y, τ) of (Y, σ) and the groupings for newly introduced vertices. We will refine (Y, σ), simplex by simplex, from the lowest dimension to the highest dimension by use of Theorem 3.32. To avoid confusion, use Γ, Γ_{1}, Γ_{2}, Γ′, etc. to denote the simplices of (Y, τ), after subdivision, and reserve the notations Δ, Δ′, Δ_{1}, etc. for the simplices of (Y, σ)—with original simplicial complex structure σ introduced in 4.7. As the induction assumption, we suppose that there are a subdivision (∂Δ, τ) of (∂Δ, σ) and the groupings of SPϕ_{z} for all vertices z ∈ Vertex(∂Δ, τ) with the following properties.

(1) If Δ′ is a proper face of (Δ, σ) (by a proper face of Δ, we mean a face Δ′ with Δ′ ⊂ ∂Δ) and z ∈ Δ′, then the grouping of Λ(Δ′), induced by the grouping of SPϕ_{z} satisfies

E_{i} ⊂ H_{i}(Δ′).

In other words, E_{i}^{Λ(Δ′)}(z) ⊂ H_{i}(Δ′).

(2) Let Γ be a simplex of (∂Δ, τ) with vertices z_{0}, z_{1}, ⋯, z_{j}. If Γ ⊂ Δ′, where Δ′ is a proper face of Δ, then

# (^{∘} E_{i}^{Λ(Δ′)}(z_{0}) ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{1}) ∩ ⋯ ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{j}))

≥ T_{i} − [MΩ + MΩ dim Δ′(dim Δ′ + 1)]
(≥ T_{i} − [MΩ + MΩ dim ∂Δ(dim ∂Δ + 1)]) .

(3) For each vertex z of (∂Δ, τ), Assumption 3.27 holds. I.e.,

#(^{∘} E_{i}^{Λ(Δ′)}(z)) ≥ T_{i} − MΩ

for any proper face Δ′ of Δ with z ∈ Δ′. (In the above conditions (1), (2) and (3), {E_{i}^{Λ(Δ′)}(z)}i=1^{L} are regarded as groupings of the set Λ(Δ′) (with multiplicity); see 4.11.) Now we define the subdivision (Δ, τ) of (Δ, σ) and the groupings for all newly introduced vertices. The restriction of the simplicial structure (Δ, τ) on ∂Δ will be the same as (∂Δ, τ), that is, we will only introduce new vertices inside interior(Δ) = Δ\∂Δ. We need to define the groupings as groupings of Λ(Δ). Then they will induce groupings of SPϕ_{z}. Claim: For any vertex z of (∂Δ, τ), if the grouping of SPϕ_{z} is regarded as the grouping of Λ(Δ), then (1’) E_{i}(z) ⊂ H_{i}(Δ) for i = 1, 2, ⋯, L. In other words, E_{i}^{Λ(Δ)}(z) ⊂ H_{i}(Δ). Proof of the Claim: Let y be the point y_{i} in the definition of Λ(Δ) in 4.9. Then Δ ⊂ O(y) ∈ O. (We avoid the notation y_{i}, since i is used for E_{i} above. So we use y instead.)

Let λ ∈ E_{i}(z). Since z ∈ ∂Δ, there is a proper face Δ′ of Δ such that z ∈ Δ′. By (1) above, E_{i}(z) ⊂ H_{i}(Δ′) (regarded as a grouping of Λ(Δ′)). From the end of 4.13, there is an index s such that λ ∈ Λ(Δ′, s)(z) and that {Λ(Δ′, s)^{∼w_{s}(Δ′)}} ⊂ H_{i}(Δ′). Recall, from 4.12, Λ(Δ′, s)(z′) is a finite union Λ(Δ, t_{1})(z′) ∪ Λ(Δ, t_{2})(z′) ∪ ⋯, for any z′ ∈ Δ′ ⊂ Δ. Hence, there is an index t such that

λ ∈ Λ(Δ, t)(z) ⊂ Λ(Δ′, s)(z),

where both Λ(Δ, t)(z) and Λ(Δ′, s)(z) are regarded as subsets of SPϕ_{z}. To prove the claim, by the end of 4.13, we only need to prove {Λ(Δ, t)^{∼w_{t}(Δ)}} ⊂ H_{i}(Δ). From the definition of H_{i}(Δ), this is equivalent to

(A)  Λ(Δ, t)(z_{1}) ⊂ \{x : dist(x, a_{i}(z_{1})) < η + \frac{dim(Δ) + 1}{(dim Y + 1)} · η\}

for any z_{1} ∈ Δ. From {Λ(Δ′, s)^{∼w_{s}(Δ′)}} ⊂ H_{i}(Δ′) and the definition of H_{i}(Δ′), we have

Λ(Δ′, s)(z′) ⊂ \{x : dist(x, a_{i}(z′)) < η + \frac{dim(Δ′) + 1}{(dim Y + 1)} · η\}

for any z′ ∈ Δ′. In the above, if we choose z′ = z—the vertex in the claim— (and note that Λ(Δ, t)(z) ⊂ Λ(Δ′, s)(z)), then

(a)  Λ(Δ, t)(z) ⊂ \{x : dist(x, a_{i}(z)) < η + \frac{dim(Δ′) + 1}{(dim Y + 1)} · η\}.

On the other hand, from (d) in 4.6, we have

(b)  diameter(a_{i}(Δ)) ≤ diameter(a_{i}(O(y))) < \frac{η}{2(dim Y + 1)}.

And from (b) in 4.6, we have

(c)  diameter(U(y, t)) < \frac{η}{2(dim Y + 1)}.

From 4.6 and 4.9, Λ(Δ, t)(z_{1}) ⊂ U(y, t) for any z_{1} ∈ Δ ⊂ O(y). Combining this with (c) above, for any μ ∈ Λ(Δ, t)(z_{1}) (z_{1} ∈ Δ), we have

dist(μ, Λ(Δ, t)(z)) < \frac{η}{2(dim Y + 1)}.

Then combining it with (a) above, we have

dist(μ, a_{i}(z)) < η + \frac{(dim(Δ′) + 1)η}{(dim Y + 1)} + \frac{η}{2(dim Y + 1)}.

Documenta Mathematica 7 (2002) 255–461

Finally, combining it with (b),

dist(μ, a_{i}(z_{1})) < η + \frac{(dim(Δ′) + 1)η}{(dim Y + 1)} + \frac{η}{2(dim Y + 1)} + \frac{η}{2(dim Y + 1)}

≤ η + \frac{dim(Δ) + 1}{(dim Y + 1)},

since dim(Δ′) ≤ dim(Δ) − 1. Note that z_{1} ∈ Δ and μ ∈ Λ(Δ, t)(z_{1}) are arbitrary, this proves (A) and the Claim. Suppose that Γ is a simplex of (∂Δ, τ) with vertices z_{0}, z_{1}, ⋯ z_{j}. Suppose that Γ ⊂ Δ′, where Δ′ is a face of Δ. As mentioned in 4.12,

∘ E_{i}^{Λ(Δ′)}(z)|_{z′} ⊂ E_{i}^{Λ(Δ)}^{∘}(z)|_{z′}

as a subset of SPϕ_{z′} for all z′ ∈ Δ′ and all z = z_{0}, z_{1}, ⋯, z_{j}. Therefore, from (2) and (3) above, we have the following (2’) and (3’).

(2’) # (E_{i}^{Λ(Δ)}^{∘}(z_{0}) ∩E_{i}^{Λ(Δ)}^{∘}(z_{1}) ∩ ⋯ ∩E_{i}^{Λ(Δ)}^{∘}(z_{j}))

= # (E_{i}^{Λ(Δ)}^{∘}(z_{0})|_{z′} ∩E_{i}^{Λ(Δ)}^{∘}(z_{1})|_{z′} ∩ ⋯ ∩E_{i}^{Λ(Δ)}^{∘}(z_{j})|_{z′})

≥ # (^{∘} E_{i}^{Λ(Δ′)}(z_{0})|_{z′} ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{1})|_{z′} ∩ ⋯ ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{j})|_{z′})

= # (^{∘} E_{i}^{Λ(Δ′)}(z_{0}) ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{1}) ∩ ⋯ ∩ ^{∘} E_{i}^{Λ(Δ′)}(z_{j}))

≥ T_{i} − [MΩ + MΩ dim ∂Δ(dim ∂Δ + 1)],

for every simplex Γ ⊂ (∂Δ, τ) with vertices z_{0}, z_{1}, ⋯, z_{j}. (3’) The Assumption 3.27 holds for the grouping {E_{i}(z)}_{i=1}^{L} regarded as a grouping of Λ(Δ), i.e.,

#(E_{i}^{Λ(Δ)}^{∘}(z)) ≥ T_{i} − MΩ,

where z is a vertex of (∂Δ, τ). Apply Theorem 3.32 to obtain a subdivision (Δ, τ) of (Δ, σ), and, for each newly introduced vertex z ∈ Δ, a grouping E_{1}(z), E_{2}(z), ⋯ E_{L}(z) of SPϕ_{z} such that (1), (2) and (3) hold with the version obtained by replacing Δ′ by Δ, and dim(∂Δ) by dim Δ. (As mentioned in 4.11, for each vertex z, we should first get the groupings of Λ(Δ), then this grouping induces a grouping of SPϕ_{z}.) Using Mathematical Induction, combined with 4.13, we obtain our subdivision (Y, τ) of (Y, σ) and the groupings. We summarize what we obtained in 4.13 and 4.14 as in the following proposition. Proposition: There is a subdivision (Y, τ) of (Y, σ), and for all vertices z ∈ Vertex(Y, τ), there are groupings E_{1}(z), E_{2}(z), ⋯, E_{L}(z) of SPϕ_{z} of type (T_{1}, T_{2}, ⋯, T_{L}) (i.e., #(E_{i}(z)) = T_{i} ∀i) such that the following are true.

(1) If Δ is a simplex of (Y, σ) (before subdivision) and z ∈ Δ, then the grouping (E_{1}^{Λ(Δ)}(z), E_{2}^{Λ(Δ)}(z), ⋯ E_{L}^{Λ(Δ)}(z)) of Λ(Δ), induced by the grouping (E_{1}(z), E_{2}(z), ⋯, E_{L}(z)) of SPϕ_{z}, satisfies

E_{i}^{Λ(Δ)}(z) ⊂ H_{i}(Δ).

(2) Let Γ be a simplex of (Y, τ) with vertices z_{0}, z_{1}, ⋯, z_{j}. If Γ ⊂ Δ, where Δ is a simplex of (Y, σ) (before subdivision), then

# (E_{i}^{Λ(Δ)}^{∘}(z_{0}) ∩E_{i}^{Λ(Δ)}^{∘}(z_{1}) ∩ ⋯ ∩E_{i}^{Λ(Δ)}^{∘}(z_{j}))

≥ T_{i} − [MΩ + MΩ dim Δ(dim Δ + 1)]
(≥ T_{i} − [MΩ + MΩ dim Y (dim Y + 1)]) .

(We do not need the condition (3) any more.)

4.15. For the simplicial complex (Y, τ), there is a finite open cover

{W (Γ) : Γ is a simplex of (Y, τ)}

of Y, with the following properties.

(a) W (Γ) ⊃ interior(Γ) = Γ\∂Γ.

(b) If W (Γ_{1}) ∩ W (Γ_{2}) ≠ ∅, then either Γ_{1} is a face of Γ_{2} or Γ_{2} is a face of Γ_{1}. (Such open cover has been constructed in 1.4.2 (b).) For any simplex Γ, we will construct an open set O(Γ) ⊃ Γ and introduce a decomposition Ξ(Γ) of {SPϕ_{y}}_{y∈O(Γ)}, which is the finest possible decomposition satisfying the condition (S) for Γ in place Y′ in 4.10. Recall that K = rank(P), and y ↦ SPϕ_{y} defines a map SPϕ : Y → P^{K}X. We will prove the following easy fact. Claim 1: SPϕ|_{Γ} := \overline{∪_{z∈Γ}SPϕ_{z}} (⊂ X) has at most K connected components. (For K = 1, the claim says that the image of a connected space Γ under a continuous map SPϕ : Γ → P^{1}X = X is connected. This is a trivial fact.) Proof of Claim 1: Suppose that by the contrary, SPϕ|_{Γ} has more than K connected components. Write SPϕ|_{Γ} = X_{1} ∪ X_{2} ∪ ⋯ ∪ X_{K+1}, where X_{1}, X_{2}, ⋯, X_{K+1} are mutually disjoint non empty closed subsets (which are not necessary connected). There are open sets U_{1}, U_{2}, ⋯, U_{K+1} with mutually disjoint closures such that U_{i} ⊃ X_{i}. Then for any z ∈ Γ, SPϕ_{z} ⊂ ∪_{i=1}^{K+1}U_{i}. By Lemma 1.2.9, for each i, #(SPϕ_{z} ∩ U_{i}) is a nonzero constant. Hence #(SPϕ_{z}) ≥ K + 1, contradicting with #(SPϕ_{z}) = K = rank(P), counting multiplicity. This proves the claim. We are back to our construction of open set O(Γ) and decomposition Ξ(Γ). Write SPϕ|_{Γ} = X_{1} ∪ X_{2} ∪ ⋯ ∪ X_{t}, where X_{1}, X_{2}, ⋯, X_{t} (with t ≤ K) are mutually disjoint connected components of SPϕ|_{Γ}. Choose open sets U_{1}, U_{2}, ⋯, U_{t} with mutually disjoint closures such that X_{i} ⊂ U_{i}. By Lemma 1.2.9, there is an open set O(Γ) ⊃ Γ such that SPϕ|_{O(Γ)} ⊂ ∪_{i=1}^{t}U_{i}. As in 4.10, define

Ξ(Γ, t)(z) = SPϕ_{z} ∩ U_{i}, ∀z ∈ O(Γ), i = 1, 2, ⋯, t.

This gives a decomposition

SPϕ_{z} = {Ξ(Γ, 1)(z), Ξ(Γ, 2)(z), ⋯, Ξ(Γ, t)(z)}, ∀z ∈ O(Γ).

Let c_{i} = #(Ξ(Γ, i), counting multiplicity. And write

Ξ(Γ) := {Ξ(Γ, 1)^{∼c_{1}}, Ξ(Γ, 2)^{∼c_{2}}, ⋯, Ξ(Γ, t)^{∼c_{t}}}.

Note that the above decomposition satisfies condition (S) in 4.10 as the decomposition of spectrum on O(Γ) (not only on Γ). In 4.16 below, when we apply 4.10, we will use U(Γ) (a subset of O(Γ)) in place of Y′ of 4.10. Obviously, Ξ(Γ) is the finest decomposition among all the decompositions of (SPϕ_{z})_{z∈Γ} satisfying condition (S) on Γ, since each X_{i} is connected. In particular, if z ∈ Γ ⊂ Δ, where Δ is a simplex of (Y, σ) (before subdivision), then the decomposition of SPϕ_{z} corresponding to Ξ(Γ) is finer than the decomposition of SPϕ_{z} corresponding to Λ(Δ). We will use the following fact later. Claim 2: If Γ′ ⊂ Γ is a face, then for any z ∈ O(Γ′) ∩ O(Γ), the decomposition of SPϕ_{z} corresponding to Ξ(Γ′) is finer than the decomposition of SPϕ_{z} corresponding to Ξ(Γ). Proof of Claim 2: The Claim follows from the definition of Ξ(Γ) and the fact that any connected component of SPϕ|_{Γ′} is completely contained in a connected component of SPϕ|_{Γ}.

4.16. For each simplex Γ, define U(Γ) = W (Γ) ∩ O(Γ). {U(Γ); Γ is a simplex of (Y, τ)} is an open covering of Y since U(Γ) ⊃ interior(Γ). For each U = U(Γ), we will define mutually orthogonal projection valued functions

P_{1}^{U}, P_{2}^{U}, ⋯, P U_{L} : U(Γ) ∋ y ↦ { sub-projections of P (y)}.

Then apply Proposition 3.2 of [DNNP] to construct the globally defined projections p_{1}, p_{2}, ⋯, p_{L} for our Theorem 4.1. (Attention: For each vertex z of (Y, τ), we have a grouping E_{1}(z), E_{2}(z), ⋯, E_{L}(z) of SPϕ_{z}. It will induce a grouping of Ξ(Γ), as in 4.10, if Γ ∋ z. In the following construction of P_{i}^{U}(y), this grouping will be used. That is, we will use the decomposition of SPϕ_{z} corresponding to Ξ(Γ). The decomposition of SPϕ_{z} corresponding to Λ(Δ) will NOT be used in the definition of P_{i}^{U}(y) at all— it is only used in the estimation of rank(P_{i}^{U}). In the definition of the grouping E_{1}(z), E_{2}(z), ⋯, E_{L}(z) of SPϕ_{z}, it involves the decomposition of SPϕ_{z} corresponding to Λ(Δ). But once it has been defined, it makes sense by itself without the decomposition of SPϕ_{z} corresponding to Λ(Δ) as a reference (though E_{i}^{Λ(Δ)}^{∘}(z)|_{z} only makes sense with the decomposition as the reference).) Back to our construction. For each y ∈ U (= U(Γ)) and each i = 1, 2, ⋯, L, define P_{i}^{U}(y) to be the spectral projection of ϕ_{y} corresponding to

(E_{i}^{Ξ(Γ)}^{∘}(z_{0}) ∩E_{i}^{Ξ(Γ)}^{∘}(z_{1}) ∩ ⋯ ∩E_{i}^{Ξ(Γ)}^{∘}(z_{j})) |_{y} ,

where z_{0}, z_{1}, ⋯, z_{j} are all vertices of Γ, and the notations E_{i}^{Ξ(Γ)}^{∘}(z) and E_{i}^{Ξ(Γ)}^{∘}(z)|_{y} are as in 4.10. (That is, E_{i}^{Ξ(Γ)}^{∘}(z) is a subset of Ξ(Γ) and E_{i}^{Ξ(Γ)}^{∘}(z)|_{y} is a subset of SPϕ_{y}.) By Lemma 1.2.9, the above functions P_{i}^{U}(y) depend on y continuously. In fact, for each i and any y ∈ U(Γ) ⊂ Y, P_{i}^{U}(y) is the spectral projection of ϕ_{y} corresponding to an open subset (of X)— in the notation of 4.15 (see the paragraph after the proof of Claim 1 in 4.15), the open subset is the union of all open subsets U_{j} ⊂ X such that

Ξ(Γ, j) ∈ E_{i}^{Ξ(Γ)}^{∘}(z_{0}) ∩E_{i}^{Ξ(Γ)}^{∘}(z_{1}) ∩ ⋯ ∩E_{i}^{Ξ(Γ)}^{∘}(z_{j})(⊂ Ξ(Γ)).

(Note that when we apply Lemma 1.2.9, we use the fact that {U_{i}} have mutually disjoint closures and SPϕ_{y} ⊂ ⋃ U_{i} from 4.15.) Recall, the decomposition of SPϕ_{z} corresponding to Ξ(Γ) is finer than any decomposition of SPϕ_{z} corresponding to Λ(Δ), if Γ ⊂ Δ. Therefore, E_{i}^{Ξ(Γ)}^{∘}(z_{0})|_{z} ⊃ E_{i}^{Λ(Δ)}^{∘}(z_{0})|_{z}, regarded as a subset of SPϕ_{z}, for any vertex z_{0} ∈ Γ and any point z ∈ Γ. By Condition (2) of the grouping (see 4.14),

rank(P_{j}^{U}) ≥ T_{j} − [MΩ + MΩ dim Y (dim Y + 1)]

for each U. The projections P_{i}^{U}, i = 1, 2, ⋯, L are mutually orthogonal, since they are spectral projections corresponding to mutually disjoint subsets of X. Let Γ′ be a face of Γ and z ∈ U(Γ) ∩ U(Γ′). By Claim 2 in 4.15, opposite to the case of decompositions corresponding to Λ(Δ) and Λ(Δ′), the decomposition of SPϕ_{z} corresponding to Ξ(Γ′) is finer than that corresponding to Ξ(Γ). Therefore,

E_{i}^{Ξ(Γ)}^{∘}(z_{0})|_{y} ⊂ ^{∘} E_{i}^{Ξ(Γ′)}(z_{0})|_{y}

for all z_{0} ∈ Vertex(Γ′, τ) ⊂ Vertex(Γ, τ). Combining it with the fact that Vertex(Γ′, τ) ⊂ Vertex(Γ, τ), we get

(  ⋂_{z_{j}∈Vertex(Γ,τ)}E_{i}^{Ξ(Γ)}^{∘}(z_{j})) |_{y} ⊂ (  ⋂_{z_{j}∈Vertex(Γ′,τ)} ^{∘} E_{i}^{Ξ(Γ′)}(z_{j})) |_{y}.

Consequently,

P_{i}^{U(Γ)}(y) ≤ P_{i}^{U(Γ′)}(y) if y ∈ U(Γ) ∩ U(Γ′).

Finally, from the condition (1) of the groupings (see the proposition in the end of 4.14) and the definition of H_{i}(Δ), we have

(E_{i}^{Ξ(Γ)}^{∘}(z)) |_{y} ⊂ \{λ; dist(λ, a_{i}(y)) < η + \frac{(dim Y +1)}{(dim Y +1)} · η = 2η\} ,

where Δ is any simplex of (Y, σ) satisfying Γ ⊂ Δ. Therefore, P_{i}^{U}(y) is the spectral projection of ϕ_{y} corresponding to a subset of

{λ; dist(λ, a_{i}(y)) ≤ 2η} ⊂ X.

We have proved the following lemma.

Lemma 4.17. There is a collection U of finitely many open sets which covers Y. For each open set U ∈ U, there are mutually orthogonal projection valued continuous functions

P_{1}^{U}, P_{2}^{U}, ⋯, P U_{L} : U ∋ y ↦ { sub-projections of P (y)}

with the following properties.

(1) If U_{1}, U_{2} ∈ U, and U_{1} ∩ U_{2} ≠ ∅, then either

P_{i}^{U_{1}}(z) ≤ P_{i}^{U_{2}}(z)

is true for all i = 1, 2, ⋯, L and all z ∈ U_{1} ∩ U_{2}, or

P_{i}^{U_{2}}(z) ≤ P_{i}^{U_{1}}(z)

is true for all i = 1, 2, ⋯, L and all z ∈ U_{1} ∩ U_{2}.

(2) rank(P_{i}^{U}(z)) ≥ T_{i} − [MΩ + MΩ dim Y (dim Y + 1)].

(3) Each P_{i}^{U}(z) is a spectral projection of ϕ_{z} corresponding to a subset of

{λ; dist(λ, a_{i}(z)) < 2η} .

4.18. For i = 1, 2, ⋯, L, applying Proposition 3.2 of [DNNP] to {P_{i}^{U}}_{U∈U}, there exist continuous projection valued functions

p_{1}^{U}, p_{2}^{U}, ⋯, pU_{L} : Y ∋ y ↦ { sub-projections of P (y)}

such that

p_{i}(y) ≤ ⋁{P_{i}^{U}(y); y ∈ U ∈ U}

and that

rank(p_{i}) ≥ T_{i} − [MΩ + MΩ dim Y (dim Y + 1)] − dim Y > T_{i} − T .

(Note that T = 2^{L}(dim X + dim Y)^{3}.)

By Condition (1) of 4.17, for each y,

span{P_{i}^{U}; y ∈ U ∈ U} = P_{i}^{U_{0}}

for a certain U_{0} ∋ y which does not depend on i. Therefore, {p_{i}(y)}i=1^{L} are mutually orthogonal since {P_{i}^{U_{0}}}i=1^{L} are mutually orthogonal.

4.19. We will prove that the above projections {p_{i}}i=1^{L} and p_{0} = P − ∑i=1^{L} p_{i} are as desired in Theorem 4.1. This is a routine calculation, as in the proof of Theorem 2.7 of [GL1] or the last part of the proof of Theorem 2.21 of [EG2]. (See 1.5.4 and 1.5.7 also.) Since we need an extra property of p_{0}ϕp_{0} (described in 4.20 below), we write down the complete proof. For each y ∈ Y, as mentioned in 4.18, there exists an open set U_{0} ∈ U with U_{0} ∋ y such that

span{P_{i}^{U}; y ∈ U ∈ U} = P_{i}^{U_{0}}, i = 1, 2, ⋯, L.

Let P_{i}(y) = P_{i}^{U_{0}}(y). Then

p_{i}(y) ≤ P_{i}(y), i = 1, 2, ⋯, L,

and each P_{i}(y) is the spectral projection corresponding to a certain subset of

{λ; λ ∈ SPϕ_{y}, dist(λ, a_{i}(y)) < 2η} .

Let mutually different elements μ_{1}, μ_{2}, ⋯, μ_{s} ∈ SPϕ_{y} be the list of spectra which are not in the set of those spectra belonging to the projections {P_{i}(y)}i=1^{L}. Let q_{1}, q_{2}, ⋯, q_{s} be spectral projections corresponding to {μ_{1}}, {μ_{2}}, ⋯, {μ_{s}}, respectively. (The rank of each q_{i} is the multiplicity of μ_{i} in SPϕ_{y}.) Then

P (y) = ∑_{i=1}^{L} P_{i}(y) + ∑_{i=1}^{s} q_{i}.

Therefore,

p_{0}(y) = P (y) − ∑_{i=1}^{L} p_{i}(y) = ∑_{i=1}^{L}(P_{i}(y) − p_{i}(y)) + ∑_{i=1}^{s} q_{i}.

Since the spectra belonging to P_{i}(y) are within distance 2η of a_{i}(y), by the way η is chosen in 4.4, for each f ∈ F,

∥ϕ(f)(y) − [∑_{i=1}^{L} f(a_{i}(y))P_{i} + ∑_{i=1}^{s} f(μ_{i})q_{i}]∥ < \frac{ε}{3}.

Therefore, for each f ∈ F, ∥p_{0}(y)ϕ(f)(y) − ϕ(f)(y)p_{0}(y)∥ < \frac{2ε}{3}, and

(∗) ∥p_{0}(y)ϕ(f)(y)p_{0}(y) − [∑_{i=1}^{L} f(a_{i}(y))(P_{i}(y) − p_{i}(y)) + ∑_{i=1}^{s} f(μ_{i})q_{i}]∥ < \frac{ε}{3}.

Also, for all f ∈ F,

∥p_{i}(y)ϕ(f)(y) − f(a_{i}(y))p_{i}(y)∥ < \frac{ε}{3} and

∥ϕ(f)(y)p_{i}(y) − f(a_{i}(y))p_{i}(y)∥ < \frac{ε}{3}.

Let P′ = ∑i=1^{L} p_{i}. Then

∥P′(y)ϕ(f)(y)p_{0}(y)∥ = ∥ ∑i=1^{L} p_{i}(y)ϕ(f)(y)p_{0}(y)∥

≤ ∥ ∑i=1^{L}[p_{i}(y)ϕ(f)(y) − p_{i}(y)f(a_{i}(y))]p_{0}(y)∥+∥ ∑i=1^{L} p_{i}(y)f(a_{i}(y))p_{0}(y)∥

≤ \frac{ε}{3} + 0 = \frac{ε}{3} .

Similarly, for all f ∈ F,

∥p_{0}(y)ϕ(f)(y)P′(y)∥ < \frac{ε}{3}.

Also,

∥P′(y)ϕ(f)(y)P′(y) − ⨁_{i=1}^{L} f(a_{i}(y))p_{i}(y)∥ < \frac{ε}{3}.

Combining all the above estimations, we have, for f ∈ F,

∥ϕ(f)(y) − p_{0}(y)ϕ(f)(y)p_{0}(y) ⊕ ⨁_{i=1}^{L} f(a_{i}(y))p_{i}(y)∥ < \frac{ε}{3} + \frac{ε}{3} + \frac{ε}{3} = ε.

This ends the proof of Theorem 4.1.

⊓ ⊔

Attention: In fact, we proved that the conclusion of Theorem 4.1 holds not only for f in the finite set F, but also for any f satisfying the condition that if dist(x, x′) < 2η, then ∥f(x) − f(x′)∥ < \frac{ε}{3}.

Remark 4.20. The following is the (∗) from 4.19:

(∗) ∥p_{0}(y)ϕ(f)(y)p_{0}(y) − [∑_{i=1}^{L} f(a_{i}(y))(P_{i}(y) − p_{i}(y)) + ∑_{i=1}^{s} f(μ_{i})q_{i}]∥ < \frac{ε}{3}.

Recall that for any x, x′ ∈ X, if dist(x, x′) < 2η, then

∥f(x) − f(x′)∥ < \frac{ε}{3}

for all f ∈ F. Note that ξ_{y} : C(X) → p_{0}(y)M_{•}(C^{|} )p_{0}(y), defined by ξ_{y}(f) = ∑i=1^{L} f(a_{i}(y))(P_{i}(y) − p_{i}(y)) + ∑_{i=1}^{s} f(μ_{i})q_{i}, is a homomorphism. By 1.2.18, we have the following claim.

Claim: Let {x_{1}, x_{2}, ⋯, x_{r}} be an η-dense subset of X. Suppose that mutually orthogonal projections p^{1}, p^{2}, ⋯, p^{r} ∈ (P − p_{0})M_{K′}(C(Y))(P − p_{0}) satisfy

rank(p^{i}) ≥ rank(p_{0}).

Let ψ : C(X) → (p_{0} ⊕ p^{1} ⊕ p^{2} ⊕ ⋯ ⊕ p^{r})M_{K′}(C(Y))(p_{0} ⊕ p^{1} ⊕ p^{2} ⊕ ⋯ ⊕ p^{r}) be the positive linear map defined by

ψ(g) = p_{0}ϕ(g)p_{0} ⊕ ∑_{i=1}^{r} g(x_{i})p^{i} ,

for all g ∈ C(X). Then ψ(F) is weakly approximately constant to within ε. This fact will be used later.

Remark 4.21. The proof of Theorem 4.1 is very long and complicated. We point out that the following direct approaches will encounter difficulties. (These discussions have appeared in §1.5.)

1. One may let P_{i}^{U}(y) be the spectral projections corresponding to the open sets

{λ; dist(λ, a_{i}(y)) < η}

and make use of Proposition 3.2 of [DNNP] to construct the projection p_{i}. The trouble is that such {p_{i}}i=1^{L} are not mutually orthogonal since P_{i}^{U} are not mutually orthogonal.

2. For each sufficiently small neighborhood U, applying the theorem about spectral multiplicity from §2, one can construct mutually orthogonal projections {P_{i}^{U}(y)}i=1^{L} with relatively large rank such that each P_{i}^{U}(y) is the spectral projection corresponding to a subset of

{λ; dist(λ, a_{i}(y)) < η}.

But one cannot guarantee that the projection associated to ⋁{P_{i}^{U}; U ∋ y} is orthogonal to the projection associated to ⋁{P_{j}^{U}; U ∋ y}, for i ≠ j. So one still can not obtain orthogonal projections {p_{i}}i=1^{L}.

3. One may try to define p_{1}, p_{2}, ⋯, p_{L}, one by one. For example, after p_{1}(y) is defined, try to choose P_{2}^{U}(y) to be orthogonal to p_{1}(y) and to be the spectral projection of a certain subset of X. Then this subset can not be chosen to be a subset of {λ; dist(λ, a_{2}(y)) < η} since some spectra may have been taken out when p_{1}(y) is defined. In fact, this subset can be chosen to be a subset of {λ; dist(λ, a_{2}(y)) < 2η}. In this way, when we define P_{i}^{U}(y), it will be a spectral projection corresponding to a subset of

{λ; dist(λ, a_{i}(y)) < i · η}.

In order for the theorem to hold, L·η needs to be small, which makes η depend on L. This is not useful at all for the application.

Remark 4.22. Note that in 4.19, when we prove that the projections {p_{i}}_{i=1}^{L} satisfy the desired conditions (i) and (ii) of Theorem 4.1, we only use the property that for any y ∈ Y, p_{i}(y), i = 1, 2, ⋯, L, are subprojections of P_{i}(y), i = 1, 2, ⋯, L, respectively. This means, (i) and (ii) of Theorem 4.1 hold for any set of projections {p′_{i}}_{i=1}^{L} with p′_{i} ≤ p_{i}, i = 1, 2, ⋯, L. So we have the freedom to replace any p_{i} by its subprojection (with suitable rank). This fact is important for the discussion below and in 4.41 and 4.44. In what follows, we will use the fact that, for the projections in M_{K′}(C(Y)) of rank at least dim(Y), cancellation always holds. That is , if three projections p, q and r in M_{•}(C(Y)) satisfy that rank(p) > dim(Y), rank(q) > dim(Y) and p⊕r is Murry von Neumann equivalent to q ⊕r, then p is Murry von Neumann equivalent to q.

(a) In fact, in 4.19, rank(p_{i}) for our projections p_{i} satisfy the stronger condition (see 4.18):

rank(p_{i}) ≥ T_{i} − [MΩ + MΩ dim Y (dim Y + 1)] − dim Y.

From Theorem 1.2 of [Hu, Chapter 8], there is a trivial projection p′_{i} < p_{i} such that

rank(p′_{i}) ≥ rank(p_{i}) − dim Y
≥ T_{i} − [MΩ + MΩ dim Y (dim Y + 1)] − 2 dim Y.

That is, rank(p′_{i}) is still larger than T_{i} − T, where T = 2^{L}(dim X + dim Y)^{3}. (In fact it is larger than T_{i} − T + 2 dim Y.) In Theorem 4.1, replacing p_{i} by p′_{i}, one makes all the projections {p_{i}}i=1^{L} trivial.

(b) Suppose that there is an i_{0} ∈ {1, 2, ⋯, L} such that T_{i_{0}} > T + dim Y. Suppose that the projections p_{1}, p_{2}, ⋯, p_{L} are trivial as in (a). In particular, suppose that rank(p_{i_{0}}) ≥ T_{i_{0}} − T + 2 dim Y as mentioned in (a). By [Hu], P ∈ M_{K′}(C(Y)) (the total projection of the target algebra P M_{K′}(C(Y))P in Theorem 4.1) can be written in the form

q ⊕ (trivial projection),

where q is of rank T_{i_{0}} − T + dim Y. It follows from [Hu], that there is a subprojection p′_{i_{0}} of p_{i_{0}} which is unitarily equivalent to q. Replacing p_{i_{0}} by p′_{i_{0}}, and keeping all the other projections p_{i}, then P will be unitarily equivalent to a projection of the form

⨁_{i=1}^{L} p_{i} ⊕ (trivial projection).

Therefore, p_{0} = P − ⨁i=1^{L} p_{i} is a trivial projection. (Note that rank(p_{0}) ≥ dim Y.) In other words, in Theorem 4.1, we can choose all the projections p_{0}, p_{1}, ⋯, p_{L} to be trivial except one of them, p_{i_{0}}, where i_{0} ≠ 0. In particular, p_{0} is a trivial projection, as comparing with (a) above.

The following theorem is proved in [EGL].

Theorem 4.23. ([EGL]) Let A = lim_{n→∞}(A_{n}, ϕ_{n,m}) be an inductive limit C^{∗}- algebra (not necessarily unital) with

A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})),

where X_{n,i} are simplicial complexs. Then one can write A = lim_{n→∞}(B_{n}, ψ_{n,m}) with

B_{n} = ⨁_{i=1}^{t_{n}} M_{{n,i}}(C(Y_{n,i})),

where Y_{n,i} are (not necessarily connected) simplicial complexs, with dim(Y_{n,i}) ≤ dim(X_{n,i}), such that all the connecting maps ψ_{n,m} are injective. Furthermore, if (A_{n}, ϕ_{n,m}) satisfies the very slow dimension growth condition, then so does (B_{n}, ψ_{n,m}).

4.24. Without loss of generality, in the rest of this article, we will assume that the connecting maps ϕ_{n,m} in the inductive limit system are injective. Without this assumption one can still prove all the theorems in this paper by modifying our arguments, and by passing to some good subsets of X_{n,i}. But this assumption makes the discussions much simpler. As mentioned in 1.1.5, we will suppose that the inductive limit algebra A = lim_{n→∞}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) satisfies the very slow dimension growth condition.

4.25. As a consequence of Theorem 4.1 and the lemma inside 1.5.11— a result due to Li—, one can obtain a decomposition for each (partial map of a) connecting map ϕ_{n,m}^{i,j} (m large enough), with the major part factoring though an interval algebra. But for our application, we need a certain part of the decomposition to be defined by point evaluations and (even if it is not large absolutely) to be relatively large compared to the “bad” part p_{0}ϕp_{0}, where p_{0} is the projection in Theorem 4.1, and ϕ is the map corresponding to ϕ_{n,m}^{i,j} (see 1.2.18 and 1.2.19), i.e., ϕ = ϕ_{n,m}^{i,j}|_{e_{11}A_{n}^{i}e_{11}}. Following Section 2 of [Li3] (see the proof of Theorem 2.28 in [Li3]), we can prove our main Decomposition Theorem (see Theorem 4.37 below). [Li3] only proves the special case that X_{n,i} = graphs (one dimensional spaces). Although the idea behind Li’s proof is reasonably simple and clear (see the explanation in 2.29 of [Li3]), the proof itself is complicated and long. It combines several difficulties together. For convenience in the higher dimensional case, we will give a slightly different approach. (See 1.5.25 for the explanation of the difference between our approach and Li’s approach.) Our proof will be a little shorter, and perhaps easier to follow (hopefully). More importantly, using this approach, we will be able to prove the Decomposition Theorem for any homomorphism provided that the homomorphism satisfies a certain quantitative condition (see Theorem 4.35 below). (Li’s theorem is for the homomorphism ϕ_{n,m} with m sufficiently large.) This slightly stronger version of the theorem is needed in [EGL] to prove the Uniqueness Theorem. It should be emphasized that our proof is essentially the same as Li’s proof in spirit. The idea behind our proof is roughly as follows. In [Li2, 2.18–2.19] (see 1.5.11), Li proves that for fixed η > 0, for m large enough, and for any (partial) connecting map ϕ_{n,m}^{i,j}—denoted by ϕ—, there are L continuous maps β_{1}, β_{2}, ⋯, β_{L} : Y (= X_{m,j}) ⟶ X (= X_{n,i}), factoring through the interval [0, 1], such that for each y ∈ Y, the set SPϕ_{y} and the set

Θ(y) = {β_{1}(y)^{∼L_{2}}, β_{2}(y)^{∼L_{2}}, ⋯, β_{L−1}(y)^{∼L_{2}}, β_{L}(y)^{∼L_{2}+L_{1}}}

can be paired within η, where L_{2} could be very large compared with L · 2^{L} · (dim(X) + dim(Y))^{3}, if the inductive limit system satisfies the very slow dimension growth condition. What we are going to prove is that, if SPϕ_{y} and Θ(y) can be paired within η, then they can still be paired within some small number (e.g., 2η), if one changes a number—a small number compared with L—of maps β_{i} to arbitrary maps (in particular, to constant maps), provided that X is path connected and ϕ has a certain spectral distribution property related to the number η and another number δ (see 4.26 below). (Note that, how many maps are allowed to be changed, also depends on η and δ.) (Those constant maps form the part of the homomorphism defined by point evaluations.) At first sight, it might seem impossible for this to be true. But, with the spectral distribution property of the homomorphism ϕ, Lemma 2.15 of [Li2] (see Lemma 4.29 below) says that if ϕ and another homomorphism ψ (in the application, ψ should be chosen to be a homomorphism with the family of spectral functions Θ(y), i.e., SPψ_{y} = Θ(y) for all y ∈ Y) are close on the level of AffT, then their spectra SPϕ_{y} and SPψ_{y} can be paired within a small number. On the other hand, changing a very few spectral functions (no matter how large a change in each function), will not create a big change on the level of AffT (see 4.28 and the claims in 4.31 below). Since the results of [Li 2] are not of a quantitative nature—they are for connecting homomorphisms ϕ_{n,m} with m large—, we can not apply them (2.18 and 2.19 of [Li 2]) directly. So we repeat part of the arguments in [Li 2]. The above method will lead us to Lemma 4.33 (see 4.26–4.33 for details). Then our main decomposition theorems—Theorem 4.35 and Theorem 4.37— will be more or less consequences. Finally we remark that, in our decomposition, we cannot require that both parts of the decomposition be homomorphisms as in 2.28 of [Li3], since in general, C(X) is not stably generated (see [Lo]).

4.26. For the reader’s convenience, we will quote some notations, terminologies and results from [Li1] and [Li2].

The following notation is inspired by a similar notation in [Li1]. For any η > 0, δ > 0, a homomorphism ϕ: P M_{k}(C(X))P → QM_{k′}(C(Y))Q is said to have the property sdp(η, δ) (spectral distribution property with respect to η and δ) if for any η−ball

B_{η}(x) := {x′ ∈ X; dist(x′, x) < η} ⊂ X

and any point y ∈ Y,

#(SPϕ_{y} ∩ B_{η}(x)) ≥ δ#(SPϕ_{y}),

counting multiplicity. (Attension: The property sdp(r, δ) in [Li1] corresponds to sdp(\frac{1}{2r} , δ) above.) Any homomorphism ϕ : ⊕ M_{k}(C(X)) → ⊕M_{l}(C(Y)) is said to have the property sdp(η, δ) if each partial map has sdp(η, δ).

4.27. The following notations can be found in Section 2 of [Li2]. Let X be a connected simplicial complex. For any closed set X_{1} ⊂ X, M > 0, let

χ_{X_{1},M}(x) = \{ \begin{matrix}1 & if x ∈ X_{1} \\ 1 − M · dist(x, X_{1}) & if dist(x, X_{1}) ≤ \frac{1}{M} \\ 0 & if dist(x, X_{1}) ≥ \frac{1}{M} . \end{matrix}

For η > 0 and δ > 0, let

H_{1}(η) = \{χ_{X_{1}, \frac{32}{η}} : X_{1} ⊂ X closed \}.

Then there is a finite set H ⊂ H_{1}(η) such that for all h ∈ H_{1}(η), dist(h, H) < \frac{δ}{8} (the distance is the distance defined by uniform norm). Denote such set by H(η, δ, X) (⊂ C(X)). Although such a set is not unique, we fix one for each triple (η, δ, X) for our purpose. (As pointed out in [Li1], the existence of such finite set H(η, δ, X) follows from equi-continuity of the functions in H_{1}(η).)

4.28. For a unital C^{∗}-algebra A, let TA denote the space of all tracial states of A, i.e., τ ∈ T A if and only if τ is a positive linear map from A to C^{|} , with τ(xy) = τ(yx) and τ(1) = 1. AffT A is the collection of all the affine maps from T A to IR. Any unital homomorphism ϕ : A → B induces an affine map

AffT ϕ : AffT A ⟶ AffT B.

It is well known, for any connected metrizable space X and any projection P ∈ M_{k}(C(X)),

AffT (P M_{k}(C(X))P) = AffT (C(X)) = CIR(X).

We would like to quote some easy facts about the AffT map from [Li1] and [Li2].

If ϕ : C(X) → P M_{l}(C(Y))P is a unital homomorphism and rank(P) = k, then AffT ϕ : C(X) → C(Y) is given by

AffT ϕ(f) = \frac{1}{k} ∑_{i=1}^{l} ϕ(f)_{ii},

where each ϕ(f)_{ii} is the diagonal entry of ϕ(f) ∈ P M_{l}(C(Y))P ⊂ M_{l}(C(Y)) at the place (i, i) For a continuous map β : Y → X, let β^{∗} : C(X) → C(Y) be defined by

β^{∗}(f) = f ∘ β (∈ C(Y)) for any f ∈ C(X).

Suppose that β_{1}, β_{2}, ⋯ β_{l} : Y → X are continuous maps. If ψ : C(X) → M_{l}(C(Y)) is a homomorphism with {β_{i}}_{i=1}^{l} as the set of spectral functions, (e.g., ψ is defined by ψ(f) = diag(β_{1}^{∗}(f), β_{2}^{∗}(f), ⋯, β_{l}^{∗}(f)),) then

AffT ψ(f) = \frac{1}{l} ∑_{i=1}^{l} β_{i}^{∗}(f).

(Let H ⊂ CIR(X) be a finite subset satisfying ∥f∥ ≤ 1 for any f ∈ H. If one modifies the above homomorphism ψ to a new homomorphism ψ′, by replacing k functions from the set of spectral functions {β_{i}}_{i=1}^{l} by other functions (from Y to X), then

∥AffT ψ(f) − AffT ψ′(f)∥ ≤ \frac{k}{l} , ∀f ∈ H.

In particular, this modification (from ψ to ψ′) does not create a big change on the level of AffT, provided that k is very small compared with l, as mentioned in 4.25.) For a unital homomorphism ϕ : C(X) → P M_{l}(C(Y))P with rank(P) = k, quoting from 1.9 of [Li1], we have

AffT ϕ(f)(y) = \frac{1}{k} ∑_{x_{i}(y)∈SPϕ_{y}} f(x_{i}(y)).

Consider e_{y} : P M_{l}(C(Y))P → P (y)M_{l}(C^{|} )P (y) (≅ M_{rank(P)}(C^{|} )), which is the homomorphism defined by evaluation at the point y. Then from the above paragraph, we know that AffT (e_{y} ∘ ϕ) depends only on SPϕ_{y}. We can denote e_{y} ∘ ϕ by ϕ|_{y} ( this is the homomorphism ϕ|_{{y}} in 1.2.13 for the single point set {y}).

Lemma 4.29. ([Li2,2.15]) Suppose that two unital homomorphisms ϕ : C(X) → P M_{k}(C(Y))P and ψ : C(X) → QM_{k}(C(Y))Q with rank(P) = rank(Q), satisfy the following two conditions:

(1) ϕ has the property sdp(\frac{η}{32}, δ);

(2) ∥AffT ϕ(h) − AffT ψ(h)∥ < \frac{δ}{4}, for all h ∈ H(η, δ, X).

Then SPϕ_{y} and SPψ_{y} can be paired within \frac{η}{4} for any y ∈ Y. (Notice that, no matter how small the δ is , the above conditions (1) and (2) do not imply the other assumption rank(P) = rank(Q), which is necessary for our conclusion.)

Proof: If P = Q, this is exactly 2.15 of [Li2]. (Notice that, we use \frac{η}{32} in place of \frac{η}{8} of [Li2 2.15], so our conclusion is that, SPϕ_{y} and SPψ_{y} can be paired within \frac{η}{4} (instead of η). Also notice that the set H in [Li2 2.15] is chosen to be the same as the above set H(η, δ, X).) To see the general case, fix y ∈ Y. We can consider two maps ϕ|_{y} and ψ|_{y} which are unital homomorphisms from C(X) to C^{∗}-algebras which are isomorphic to the same C^{∗}-algebra M_{rank(P)}(C^{|} ). (Note that rank(P) = rank(Q).) The conditions (1) and (2) above imply the same conditions for ϕ|_{y} and ψ|_{y}, since AffT ϕ(h)(y) = AffT (ϕ|_{y})(h). Therefore, by 2.15 of [Li 2], SPϕ_{y} = SP (ϕ|_{y}) and SPψ_{y} = SP (ψ|_{y}) can be paired within \frac{η}{4} . (If one checks the proof of 2.15 of [Li2] carefully, then he will easily recognize that the above Lemma is already proved there.)

⊓ ⊔

4.30. In the following paragraphs (4.30—4.32), we will apply the materials from 2.8 – 2.10 of [Li2]. For any η > 0 and δ > 0, from 2.9 of [Li2], there exist a continuous map α : [0, 1] → X, and a unital positive linear map ξ : C[0, 1] → C(X) such that

∥ξ∘α^{∗}(f) − f∥ < \frac{δ}{16},

for each f ∈ H(η, δ, X), where α^{∗} : C(X) → C[0, 1] is induced by α. Furthermore, we can choose α such that image(α) is \frac{η}{32}-dense in X. For α : [0, 1] → X, there is a σ > 0 such that |t − t′| < 2σ implies that

dist(α(t), α(t′)) < \frac{η}{32}.

For a fixed space X, the number σ depends only on η and δ, since so does the continuous map α. We denote the number σ by σ(η, δ).

4.31. Let H̃ = α^{∗}(H(η, δ, X)) ⊂ C[0, 1]. For the finite set H̃ and \frac{δ}{16} > 0, there is an integer N (as in Theorem 2.1 of [Li2]) such that for any positive linear map ζ : C[0, 1] → C(Y), and for any r ≥ N, there are r continuous maps

β_{1}, β_{2}, ⋯, β_{r} : Y ⟶ [0, 1]

such that

∥ζ(f) − \frac{1}{r} ∑_{i=1}^{r} β_{i}^{∗}(f)∥ < \frac{δ}{16}

for all f ∈ H̃, where β_{i}^{∗} : C[0, 1] → C(Y) is induced by β_{i}.

We will also assume \frac{1}{N} < \frac{δ}{64}. Then we can prove the following claim. Claim 1: For any r ≥ N, if

∥ζ(f) − \frac{1}{r} ∑_{i=1}^{r} β_{i}^{∗}(f)∥ < \frac{δ}{16} for all f ∈ H̃,

then for any other two continuous maps τ_{1}, τ_{2} : Y ⟶ [0, 1],

∥ζ(f) − \frac{1}{r + 2} (∑_{i=1}^{r} β_{i}^{∗}(f) + τ_{1}^{∗}(f) + τ_{2}^{∗}(f))∥ < \frac{δ}{8}, ∀f ∈ H̃

Proof of the claim: The claim follows from

∥ζ(f) − \frac{1}{r + 2} (∑_{i=1}^{r} β_{i}^{∗}(f) + τ_{1}^{∗}(f) + τ_{2}^{∗}(f))∥

≤ ∥ζ(f) − \frac{1}{r} ∑_{i=1}^{r} β_{i}^{∗}(f)∥ + ∥\frac{1}{r} ∑_{i=1}^{r} β_{i}^{∗}(f) − \frac{1}{r + 2} ∑_{i=1}^{r} β_{i}^{∗}(f)∥

+ ∥\frac{1}{r + 2}(τ_{1}^{∗}(f) + τ_{2}^{∗}(f))∥

< \frac{δ}{16} + 2 · \frac{δ}{64} + 2 · \frac{δ}{64} = \frac{δ}{8}

for any f ∈ H̃. In the above estimation, we use the facts that ∥f∥ ≤ 1, ∥β_{i}^{∗}(f)∥ ≤ 1 and ∥τ_{i}^{∗}(f)∥ ≤ 1 for any f ∈ H̃. In the above claim, if we replace the condition r ≥ N by the condition r ≥ mN, then in the conclusion, we can allow 2m continuous maps τ_{1}, τ_{2}, ⋯, τ_{2m} : Y ⟶ [0, 1], instead of two maps. Namely, the following claim can be proved in exactly the same way. Claim 2: For any r ≥ mN, if

∥ζ(f) − \frac{1}{r} ∑_{i=1}^{r} β_{i}^{∗}(f)∥ < \frac{δ}{16}, ∀ f ∈ H̃,

then for any 2m continuous maps τ_{1}, τ_{2}, ⋯, τ_{2m} : Y ⟶ [0, 1],

∥ζ(f) − \frac{1}{r + 2m} (∑_{i=1}^{r} β_{i}^{∗}(f) + ∑_{i=1}^{2m} τ_{i}^{∗}(f))∥ < \frac{δ}{8}, ∀f ∈ H̃.

4.32. Let n = int (\frac{1}{σ(η,δ)}) + 1, where int(·) denote the integer part of the number (see 1.1.7 (c)). Divide [0, 1] into n intervals such that each of them has length at most σ(η, δ). Choose n points

t_{1}, t_{2}, ⋯, t_{n},

one from each of the intervals. Let x_{i} = α(t_{i}) ∈ X, i = 1, 2, ⋯, m. Then the set

{x_{1}, x_{2}, ⋯, x_{n}}

is \frac{η}{16}-dense in X by the way σ is chosen in 4.30. From the above discussion, for fixed η > 0, δ > 0, and the space X, we can find α, ξ, σ, N, n, H(η, δ, X), H̃, the set \{t_{1}, t_{2}, ⋯, t_{n}\} ⊂ [0, 1], and the \frac{η}{16}-dense set {x_{1}, x_{2}, ⋯, x_{n}} ⊂ X. All of them depend only on η, δ, and the space X.

Lemma 4.33. For any connected simplicial complex X, any numbers η > 0 and δ > 0, there are integers n, N, a continuous map α : [0, 1] → X, and finitely many points \{t_{1}, t_{2}, ⋯, t_{n}\} ⊂ [0, 1] with \{α(t_{1}), α(t_{2}), ⋯, α(t_{n})\} \frac{η}{16}-dense in X, such that the following is true. (Denote L := n(N + 2).) If a unital homomorphism ϕ : C(X) → P M_{k}(C(Y))P satisfies the following two conditions:

(i) ϕ has the property sdp(\frac{η}{32}, δ); (ii) rankϕ(1) := K ≥ L^{2} = (n(N + 2))^{2}, and write K = LL_{2} +L_{1} with L_{2} = int(\frac{K}{L}) and 0 ≤ L_{1} < L, (note that L ≤ L_{2}, since K ≥ L^{2},) then there are L continuous functions

β_{1}, β_{2}, ⋯, β_{n}, β_{n+1}, ⋯, β_{L} : Y ⟶ [0, 1]

such that

(1) β_{i}(y) = t_{i} for 1 ≤ i ≤ n;

(2) For each y ∈ Y, SPϕ_{y} and the set

Θ(y) = {α∘β_{1}(y)^{∼L_{2}}, α∘β_{2}(y)^{∼L_{2}}, ⋯, α∘β_{L−1}(y)^{∼L_{2}}, α∘β_{L}(y)^{∼L_{2}+L_{1}}}

can be paired within \frac{η}{2} .

(3) If Y is a connected finite simplicial complex and Y ≠ {pt}, then the map β_{n+1} : Y → [0, 1]—the first nonconstant map above—, is a surjection. (This lemma is similar to Lemma 2.18 of [Li2], but we require some of the functions β_{i} (1 ≤ i ≤ n) to be constant functions.) (Attention: To apply Theorem 4.1, one only needs SPϕ_{y} and Θ(y) to be paired within η. The advantage of using \frac{η}{2} is the following. If ψ is another homomorphism such that SPψ_{y} and SPϕ_{y} can be paired within \frac{η}{2} for any y, then we can apply Theorem 4.1 to both ϕ and ψ without requiring ψ to have the property sdp(\frac{η}{32}, δ). This observation will not be used in the proof of the main theorem of this paper. But it will be used in the proof of the Uniqueness Theorem in [EGL] (part II of the series), see 4.41–4.48 below.)

Proof: Follow the notations in 4.26 – 4.32. Let

ζ : C[0, 1] ⟶ C(Y)

be defined by ζ = AffT ϕ∘ξ. Since K − 2nL_{2} ≥ nL_{2}N, there are K − 2nL_{2} continuous maps

γ_{1}, γ_{2}, ⋯, γ_{K−2nL_{2}} : Y ⟶ [0, 1]

such that

∥ζ(f) − \frac{1}{K − 2nL_{2}} ∑_{i=1}^{K−2nL_{2}} γ_{i}^{∗}(f)∥ < \frac{δ}{16}

for all f ∈ H̃. Let

β_{1}, β_{2}, ⋯, β_{n} : Y ⟶ [0, 1]

be defined by β_{i}(y) = t_{i}. Then by Claim 2 of 4.31 (taking m = nL_{2}),

∥ζ(f) − \frac{1}{K} ( ∑_{i=1}^{K−2nL_{2}} γ_{i}^{∗}(f) + 2L_{2} ∑_{i=1}^{n} β_{i}^{∗}(f))∥ < \frac{δ}{8}

for all f ∈ H̃ ⊂ C[0, 1]. Therefore,

∥(ζ ∘α^{∗})(f) − \frac{1}{K} ( ∑_{i=1}^{K−2nL_{2}}(α∘γ_{i})^{∗}(f) + 2L_{2} ∑_{i=1}^{n}(α∘β_{i})^{∗}(f))∥ < \frac{δ}{8}

for all f ∈ H(η, δ, X). On the other hand, by 4.30 and ζ = AffT ϕ∘ξ,

∥AffT ϕ(f) − (ζ ∘α^{∗})(f)∥ < \frac{δ}{16} for f ∈ H(η, δ, X).

One can define a unital homomorphism ψ : C(X) → M_{K}(C(Y)) with {α∘γ_{i}}_{i=1}^{K−2nL_{2}} ∪ {(α∘β_{i})^{∼2L_{2}}}_{i=1}^{n} as the family of the spectral functions. Then from 4.28,

AffT ψ(f) = \frac{1}{K} ( ∑_{i=1}^{K−2nL_{2}}(α∘γ_{i})^{∗}(f) + 2L_{2} ∑_{i=1}^{n}(α∘β_{i})^{∗}(f)) .

Hence,

∥AffT ϕ(f) − AffT ψ(f)∥ ≤ \frac{δ}{8} + \frac{δ}{16} < \frac{δ}{4}

for all f ∈ H(η, δ, X). Note that rank(P) = K. By Lemma 4.29, SPϕ_{y} and

SPψ_{y} =

{α∘β_{1}(y)^{∼2L_{2}}, α∘β_{2}(y)^{∼2L_{2}}, ⋯, α∘β_{n}(y)^{∼2L_{2}}, α∘γ_{1}(y), ⋯, α∘γ_{L−2nL_{2}}(y)}

can be paired within \frac{η}{4} . Note that in our lemma, we only need L_{2} copies of each constant maps β_{i} (i = 1, 2, ⋯, n). One may wonder why we put 2L_{2} copies of each of maps β_{i} in the above set. The reason is that, after taking out L_{2} copies of β_{i}, we still want the set Θ(y) to have enough elements in each small interval of length σ, and the other L_{2} copies of β_{i} can serve for this purpose.

Consider the following set (of K − nL_{2} elements)

{β_{1}^{∼L_{2}}(y), β_{2}^{∼L_{2}}(y), ⋯, β_{n}^{∼L_{2}}(y), γ_{1}(y), γ_{2}(y), ⋯, γ_{K−2nL_{2}}(y)}.

In each interval of [0, 1] of length σ, there are at least L_{2} points (counting multiplicities) in the above set. The following argument appeared in 2.18 of [Li2]. For each fixed y, we can rearrange all the elements in the above set in the increasing order. I.e., write them as γ′_{1}(y), γ′_{2}(y), ⋯, γ′_{K−nL_{2}}(y) such that for each fixed y

{γ′_{1}(y), γ′_{2}(y), ⋯, γ′_{K−nL_{2}}(y)} =

= {β_{1}(y)^{∼L_{2}}, β_{2}(y)^{∼L_{2}}, ⋯, β_{n}(y)^{∼L_{2}}, γ_{1}(y), γ_{2}(y), ⋯, γ_{K−2nL_{2}}(y)}

(as a set with multiplicity), and such that

0 ≤ γ′_{1}(y) ≤ γ′_{2}(y) ≤ ⋯ ≤ γ′_{K−nL_{2}}(y) ≤ 1.

It is easy to prove that γ′_{i}(y), 1 ≤ i ≤ K − nL_{2} are continuous (real-valued) functions, using the following well known fact repeatedly: For any two realvalued continuous functions f and g, the functions max(f, g) and min(f, g) are also continuous. We can put each group of L_{2} consecutive functions of {γ′_{i}} (beginning with smallest one) together except the last L_{2} + L_{1} functions which will be put into a single group—the last group. Then we replace all the functions in a same group by the smallest function in the group. Namely, let

β_{n+1} = γ′_{1}, β_{n+2} = γ′_{L_{2}+1}, ⋯, β_{L} = γ′_{(L−n−1)L_{2}+1} .

Then from the fact that in each interval of [0, 1] of length σ, there are at least L_{2} points (counting multiplicity) in the set {γ′_{1}(y), γ′_{2}(y), ⋯, γ′_{K−nL_{2}}(y)}, we know that {γ′_{1}(y), γ′_{2}(y), ⋯, γ′_{K−nL_{2}}(y)} and

{β_{n+1}(y)^{∼L_{2}}, β_{n+2}(y)^{∼L_{2}}, ⋯, β_{L−1}(y)^{∼L_{2}}, β_{L}(y)^{∼L_{2}+L_{1}}}

can be paired within 2σ. Recall that |t−t′| < 2σ implies that dist(α(t), α(t′)) < \frac{η}{16}. Hence

{α∘β_{1}(y)^{∼2L_{2}}, α∘β_{2}(y)^{∼2L_{2}}, ⋯, α∘β_{n}(y)^{∼2L_{2}}, α∘γ_{1}(y), α∘γ_{2}(y), ⋯, α∘γ_{L−2nL_{2}}(y)}

and

Θ(y) = {α∘β_{1}(y)^{∼L_{2}}, α∘β_{2}(y)^{∼L_{2}}, ⋯, α∘β_{n}(y)^{∼L_{2}},

α∘β_{n+1}(y)^{∼L_{2}}, ⋯, α∘β_{L−1}(y)^{∼L_{2}}, α∘β_{L}(y)^{∼L_{2}+L_{1}}} can be paired within \frac{η}{16}. Therefore, SPϕ_{y} and Θ(y) can be paired within \frac{η}{4} + \frac{η}{16} < \frac{5η}{16} . Note that \{α∘β_{1}(y), α∘β_{2}(y), ⋯, α∘β_{n}(y)\} is \frac{η}{16} dense in X. From the proof of Lemma 1.2.17, if we replace only one map (say β_{n+1}) by an arbitrary map from Y to [0, 1], then the new Θ(y) can be paired with the old Θ(y) to within \frac{η}{8} . As a consequence, we still have that SPϕ_{y} and the new Θ(y) can be paired within \frac{5η}{16} + \frac{η}{8} < \frac{η}{2} . In particular, if Y is a connected finite simplicial complex which is not a single point, then β_{n+1} could be chosen to be a surjection, again using the Peano curve.

⊓ ⊔

4.34. Fix a large positive integer J. We require that the decomposition in 4.1 to satisfy the condition

J · (rank(p_{0}) + 2 dim(Y)) ≤ rank(p_{i}), ∀i ≥ 1.

To do so, we need rank(ϕ(1)) to be large enough. We describe it as follows. For a connected simplicial complex X, and for numbers η > 0 and δ > 0, let N, n and α : [0, 1] → X be as in Lemma 4.33. Let L = n(N + 2). Suppose that ϕ : C(X) → M_{k}(C(Y)) is a homomorphism. If ϕ has the property sdp(\frac{η}{32}, δ) and

rank(ϕ(1)) ≥ 2JL^{2} · 2^{L}(dim X + dim Y + 1)^{3},

then there are continuous functions

β_{1}, β_{2}, ⋯, β_{L} : Y ⟶ [0, 1]

(as in Lemma 4.33) such that SPϕ_{y} and the set

{α∘β_{1}(y)^{∼L_{2}}, α∘β_{2}(y)^{∼L_{2}}, ⋯, α∘β_{L−1}(y)^{∼L_{2}}, α∘β_{L}(y)^{∼L_{2}+L_{1}}}

can be paired within \frac{η}{2} , where

L_{2} = int (\frac{rankϕ(1)}{L}) ≥ 2JL · 2^{L}(dim X + dim Y + 1)^{3},

and 0 ≤ L_{1} < L. For any given set F ⊂ C(X), if η is chosen as in Theorem 4.1 (see 4.4), then by Theorem 4.1, there are mutually orthogonal projections

p_{1}, p_{2}, ⋯, p_{L} and p_{0} = ϕ(1) − ∑i=1^{L} p_{i} such that

(1) For all f ∈ F and y ∈ Y,

∥ϕ(f)(y) − p_{0}(y)ϕ(f)(y)p_{0}(y) ⊕ ⨁_{i=1}^{L} f(a_{i}(y))p_{i}(y)∥ < ε;

(2) For each i = 1, 2, ⋯, L, rank(p_{i}) ≥ L_{2} − 2^{L}(dim X + dim Y + 1)^{3}, and

J(rank(p_{0}) + 2 dim(Y)) ≤ J(L · 2^{L}(dim X + dim Y + 1)^{3} + 2 dim Y) ≤ rank(p_{i}).

By [Hu], \underbrace{p_{0} ⊕ p_{0} ⊕ ⋯ ⊕ p_{0}}_{J} is (unitarily) equivalent to a subprojection of p_{i},

since every complex vector bundle (over Y) of dimension J · rank(p_{0}) is a subbundle of any vector bundle (over Y) of dimension at least J·rank(p_{0})+dim(Y). We denote this fact by J[p_{0}] < [p_{i}]. Let Q_{0} = p_{0}, Q_{1} = p_{1} + p_{2} + ⋯ + p_{n} and Q_{2} = p_{n+1} + p_{n+2} + ⋯ + p_{L}. Then

ϕ(1) = Q_{0} + Q_{1} + Q_{2}.

Let ϕ_{0} : C(X) → Q_{0}M_{k}(C(Y))Q_{0}, ϕ_{1} : C(X) → Q_{1}M_{k}(C(Y))Q_{1} and ϕ_{2} : C(X) → Q_{2}M_{k}(C(Y))Q_{2} be defined by

ϕ_{0}(f)(y) = p_{0}ϕ(f)(y)p_{0},

ϕ_{1}(f) = ∑_{i=1}^{n} f(α∘β_{i}(y))p_{i}, and

ϕ_{2}(f)(y) = ∑_{i=n+1}^{L} f(α∘β_{i}(y))p_{i}.

Then we have the following facts.

(a) ϕ_{2} is a homomorphism factoring through C[0, 1] as

ϕ_{2} : C(X) ⟶^{ξ_{1}} C[0, 1] ⟶^{ξ_{2}} Q_{2}M_{k}(C(Y))Q_{2}.

Furthermore, if Y ≠ {pt}, then ξ_{2} is injective. (This follows from the surjection of β_{n+1}.)

(b) Note that

α∘β_{1}(y) = x_{1}, α∘β_{2}(y) = x_{2}, ⋯, α∘β_{n}(y) = x_{n}

are n constant maps with {x_{1}, x_{2}, ⋯, x_{n}} η-dense in X. By the claim in 4.20, (ϕ_{0} ⊕ ϕ_{1})(F) is approximately constant to within ε. (Note that ϕ_{0} is not a homomorphism, it is a completely positive linear ∗-contraction.) Furthermore, if η < ε, then the set {x_{1}, x_{2}, ⋯, x_{n}} is ε-dense in X. Therefore, we have proved the following theorem.

Theorem 4.35. Let X be a connected finite simplicial complex, and ε > η > 0. For any δ > 0, there is an integer L > 0 such that the following holds. Suppose that F ⊂ C(X) is a finite set such that dist(x, x′) < 2η implies |f(x)− f(x′)| < \frac{ε}{3} for all f ∈ F. If ϕ : C(X) → M_{k}(C(Y)) is a homomorphism with the property sdp(\frac{η}{32}, δ), and rank(ϕ(1)) ≥ 2J · L^{2} · 2^{L}(dim X + dim Y + 1)^{3}, where Y is a connected finite simplicial complex and J is any fixed positive integer, then there are three mutually orthogonal projections Q_{0}, Q_{1}, Q_{2} ∈ M_{k}(C(Y)), a map ϕ_{0} ∈ Map(C(X), Q_{0}M_{k}(C(Y))Q_{0})_{1} and two homomorphisms ϕ_{1} ∈ Hom(C(X), Q_{1}M_{k}(C(Y))Q_{1})_{1} and ϕ_{2} ∈ Hom(C(X), Q_{2}M_{k}(C(Y))Q_{2})_{1} such that

(1) ϕ(1) = Q_{0} + Q_{1} + Q_{2};

(2) ∥ϕ(f) − ϕ_{0}(f) ⊕ ϕ_{1}(f) ⊕ ϕ_{2}(f)∥ < ε for all f ∈ F;

(3) The homomorphism ϕ_{2} factors through C[0, 1] as

ϕ_{2} : C(X) ⟶^{ξ_{1}} C[0, 1] ⟶^{ξ_{2}} Q_{2}M_{k}(C(Y))Q_{2}.

Furthermore, if Y ≠ {pt}, then ξ_{2} is injective;

(4) The set (ϕ_{0} ⊕ ϕ_{1})(F) is approximately constant to within ε;

(5) Q_{1} = p_{1} + ⋯ + p_{n}, with J[Q_{0}] ≤ [p_{i}] (i = 1, 2 ⋯ n), ϕ_{0} is defined by ϕ_{0}(f) = Q_{0}ϕ(f)Q_{0}, and ϕ_{1} is defined by

ϕ_{1}(f) = ∑_{i=1}^{n} f(x_{i})p_{i}, ∀f ∈ C(X),

where p_{0}, p_{1}, ⋯ p_{n} are mutually orthogonal projections and {x_{1}, x_{2}, ⋯ x_{n}} ⊂ X is an ε-dense subset of X. (Again by J[p] ≤ [q], we mean that \underbrace{p ⊕ p ⊕ ⋯ ⊕ p}_{J}

is (unitarily) equivalent to a subprojection of q.) Furthermore, we can choose any two of projections Q_{0}, Q_{1}, Q_{2} to be trivial, if we wish. If ϕ(1) is trivial, then all of them can be chosen to be trivial projections. (This is remark 4.22.)

4.36. Let a simple C^{∗}-algebra A be an inductive limit of matrix algebras over simplicial complexes (A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) with injective homomorphisms. Suppose that this inductive limit system possesses the very slow dimension growth condition. In what follows, we will use the material from 1.2.19. Fix A_{n}, finite set F_{n} = ⨁_{i=1}^{t_{n}} F_{n}^{i} ⊂ A_{n}, and ε > 0. Let ε′ = \frac{ε}{max_{1≤i≤tn} \{[n,i]\}}.

Let F′^{i} ⊂ C(X_{n,i}) be the finite set consisting of all the entries of elements in F_{n}^{i} (⊂ M_{[n,i]}(C(X_{n,i}))). Let η > 0 (η ≤ ε) be such that if x, x′ ∈ X_{n,i} (i = 1, 2 ⋯ t_{n}) and dist(x, x′) < 2η, then |f(x) − f(x′)| < \frac{ε′}{3} for any f ∈ F′^{i}. For the above η > 0, there is a δ > 0 such that for sufficiently large m, each partial map ϕ_{n,m}^{i,j} : A_{n}^{i} → A_{m}^{j} has the property sdp(\frac{η}{32}, δ). (This is a consequence of simplicity of the algebra A and injectivity of ϕ_{n,m}. See [DNNP], [Ell], [Li1-2] for details.) For these numbers η and δ, and the simplicial complexes X_{n,i}, there are L(i), i = 1, 2, ⋯, t_{n}, as in Theorem 4.35. (Note that the numbers L_{i} only depend on η, δ and the spaces.) Let L = max_{i} L(i). Fix a positive integer J. By the very slow dimension growth condition, there is an integer M such that for any m ≥ M,

\frac{rankϕ_{n,m}^{i,j}(1_{A_{n}^{i}})}{rank(1_{A_{n}^{i}})} > 2J · L^{2} · 2^{L}(dim X_{n,i} + dim X_{m,j} + 1)^{3}.

Documenta Mathematica 7 (2002) 255–461

As in 1.2.16, (also see 1.2.19) each partial map ϕ_{n,m}^{i,j} : A_{n}^{i} → ϕ_{n,m}^{i,j}(1_{A_{n}^{i}})A_{m}^{j}ϕ_{n,m}^{i,j}(1_{A_{n}^{i}}) can be written as ϕ′ ⊗ 1_{[n,k]} for some homomorphism ϕ′ : C(X_{n,i}) → EA_{m}^{j}E, where E = ϕ_{n,m}^{i,j}(e_{11}), and e_{11} is the canonical matrix unit corresponding to the upper left corner. The map ϕ′ also has the property sdp(\frac{η}{32}, δ). Applying Theorem 4.35 to F′^{i} ⊂ C(X_{n,i}), η, δ, and ϕ′ (as the above) and using 1.2.19, one can obtain the following Theorem.

Theorem 4.37. For any A_{n}, finite set F = ⨁_{i=1}^{t_{n}} F^{i} ⊂ A_{n}, positive integer J, and number ε > 0, there are an A_{m}, mutually orthogonal projections Q_{0}, Q_{1}, Q_{2} ∈ A_{m} with Q_{0} + Q_{1} + Q_{2} = ϕ_{n,m}(1_{A_{n}}), a unital map ψ_{0} ∈ Map(A_{n}, Q_{0}A_{m}Q_{0})_{1}, and unital homomorphisms ψ_{1} ∈ Hom(A_{n}, Q_{1}A_{m}Q_{1})_{1}, ψ_{2} ∈ Hom(A_{n}, Q_{2}A_{m}Q_{2})_{1}, such that

(1) ∥ϕ_{n,m}(f) − ψ_{0}(f) ⊕ ψ_{1}(f) ⊕ ψ_{2}(f)∥ < ε for all f ∈ F;

(2) The set (ψ_{0} ⊕ ψ_{1})(F) is weakly approximately constant to within ε;

(3) The homomorphism ψ_{2} factors through ⨁_{i=1}^{t_{n}} M_{[n,i]}(C[0, 1]) as

ψ_{2} : A_{n} ⟶^{ξ_{1}} ⨁_{i=1}^{t_{n}} M_{[n,i]}(C[0, 1]) ⟶^{ξ_{2}} Q_{2}A_{m}Q_{2},

and ξ_{2} satisfies the following condition: if X_{m,j} ≠ {pt}, then ξ_{2}^{i,j} : M_{[n,i]}(C[0, 1]) → A_{m}^{j} is injective;

(4) Each partial map ψ_{0}^{i,j} : A_{n}^{i} → Q_{0}^{i,j}A_{m}^{j}Q_{0}^{i,j} (where Q_{0}^{i,j} = ψ_{0}^{i,j}(1_{A_{n}^{i}})) is of the form ψ′_{0} ⊗ id_{[n,k]} with ψ′_{0} : C(X_{n,i}) → q_{0}A_{m}^{j}q_{0} (where q_{0} = ψ_{0}^{i,j}(e_{11}) is a projection). Each partial map ψ_{1}^{i,j} : A_{n}^{i} → Q_{1}^{i,j}A_{m}^{j}Q_{1}^{i,j} (where Q_{1}^{i,j} = ψ_{1}^{i,j}(1_{A_{n}^{i}})) is of the form ψ′_{1} ⊗ id_{[n,k]} and ψ′_{1} : A_{n}^{i} → p^{i,j}A_{m}^{j}p^{i,j} (where p^{i,j} = ψ_{1}^{i,j}(e_{11})), satisfies the following

ψ′_{1}(f) = ∑_{i=1}^{n} f(x_{i})p_{i}

for any f ∈ C(X_{n,i}), where p_{1}, ⋯, p_{n} are mutually orthogonal projections with p^{i,j} = p_{1} +⋯+p_{n}, and with J ·[q_{0}] ≤ [p_{s}] (s = 1, 2 ⋯ n) and {x_{1}, x_{2}, ⋯ x_{n}} ⊂ X_{n,i} is an ε-dense subset in X_{n,i}. (When we apply this theorem in Section 6, Q_{0} + Q_{1} will be chosen to be a trivial projection.)

Definition 4.38. Let A = P M_{l}(C(X))P, and L be a positive integer and η > 0. A homomorphism λ : A → B = QM_{l_{1}}(C(Y))Q is said to be defined by point evaluations of size at least L at an η-dense subset if there are mutually orthogonal projections Q_{1}, Q_{2}, ⋯, Q_{n} with rank(Q_{i}) ≥ L, an η-dense subset {x_{1}, x_{2}, ⋯, x_{n}} ⊂ X, and unital homomorphisms λ_{i} : A → Q_{i}BQ_{i}, i = 1, 2, ⋯, n such that

(1) λ(1) = ∑_{i=1}^{n} Q_{i}, and λ = ⨁_{i=1}^{n} λ_{i}; (2) The homomorphisms λ_{i} factor through P (x_{i})M_{l}(C^{|} )P (x_{i}) (≅ M_{rank(P)}(C^{|} )) as

λ_{i} = λ′_{i} ∘ e_{x_{i}} : P M_{l}(C(X))P ⟶^{e_{x_{i}}}P (x_{i})M_{l}(C^{|} )P (x_{i})⟶^{λ′_{i}}Q_{i}BQ_{i},

where e_{x_{i}} are evaluation maps defined by e_{x_{i}}(f) = f(x_{i}) and λ′_{i} ∈ Hom(M_{rank(P)}(C^{|} ), Q_{i}BQ_{i})_{1}. We will also call the above homomorphism λ to have the property PE(L, η). (PE stands for point evaluation.) A homomorphism λ : A → B = QM_{l_{1}}(C(Y))Q is said to contain a part of point evaluation at point x of size at least L, if λ = λ_{1} ⊕ λ′, where λ_{1} factor through P (x)M_{l}(C^{|} )P (x) as

λ_{1} = λ′_{1} ∘ e_{x} : P M_{l}(C(X))P ⟶^{e_{x}}P (x)M_{l}(C^{|} )P (x)⟶^{λ′_{1}}Q_{1}BQ_{1},

and λ′_{1} is a unital homomorphism with rank(Q_{1}) ≥ L. The following result is a corollary of Theorem 4.37, and will also be used in the proof of our main reduction theorem.

Corollary 4.39. For any A_{n}, finite set F = ⨁_{i=1}^{t_{n}} F^{i} ⊂ A_{n}, positive integer J, any numbers ε > 0 and η > 0, and any projection P = ⊕P^{i} ∈ ⊕A_{n}^{i}, there are an A_{m}, mutually orthogonal projections Q_{0}, Q_{1}, Q_{2} ∈ A_{m} with Q_{0} +Q_{1} +Q_{2} = ϕ_{n,m}(1_{A_{n}}), a unital map ψ_{0} ∈ Map(A_{n}, Q_{0}A_{m}Q_{0})_{1}, and unital homomorphisms ψ_{1} ∈ Hom(A_{n}, Q_{1}A_{m}Q_{1})_{1}, ψ_{2} ∈ Hom(A_{n}, Q_{2}A_{m}Q_{2})_{1}, such that Part I:

(1) ∥ϕ_{n,m}(f) − ψ_{0}(f) ⊕ ψ_{1}(f) ⊕ ψ_{2}(f)∥ < ε for all f ∈ F;

(2) The homomorphism ψ_{2} factors through a direct sum of matrix algebras over C[0, 1] as

ψ_{2} : A_{n} ⟶^{ξ_{1}} ⨁_{i=1}^{t_{n}} M_{[n,i]}(C[0, 1]) ⟶^{ξ_{2}} Q_{2}A_{m}Q_{2},

and ξ_{2} satisfies the condition that, if X_{m,j} ≠ {pt}, then ξ_{2}^{i,j} : M_{[n,i]}(C[0, 1]) → A_{m}^{j} is injective.

(3) For any blocks A_{n}^{i} ⊂ A_{n}, A_{m}^{j} ⊂ A_{m}, and for the partial maps ψ_{0}^{i,j} and ψ_{1}^{i,j}, we have that ψ_{0}^{i,j}(1_{A_{n}^{i}}) := Q_{0}^{i,j} is a projection and ψ_{1}^{i,j} has the property PE(J · rank(Q_{0}^{i,j}), η).

(4) The set (ψ_{0} ⊕ ψ_{1})(F) is weakly approximately constant to within ε. Part II: ψ_{0}^{i,j}(P^{i}) and ψ_{0}^{i,j}(1_{A_{n}^{i}} − P^{i}) are mutually orthogonal projections, and the decomposition of ϕ′_{n,m} := ϕ_{n,m}|_{P A_{n}P} as the direct sum of ψ′_{0} := ψ_{0}|_{P A_{n}P}, ψ′_{1} := ψ_{1}|_{P A_{n}P}, and ψ′_{2} := ψ_{2}|_{P A_{n}P} satisfies the following conditions:

(1) ∥ϕ′_{n,m}(f) − ψ′_{0}(f) ⊕ ψ′_{1}(f) ⊕ ψ′_{2}(f)∥ < ε for all f ∈ P F P = ⊕P^{i}F^{i}P^{i};

(2) The homomorphism ψ′_{2} factors through a C^{∗}-algebra C which is a direct sum of matrix algebras over C[0, 1] as

ψ′_{2} : P A_{n}P ⟶^{ξ′_{1}} C ⟶^{ξ′_{2}} Q′_{2}A_{m}Q′_{2},

and ξ′_{2} satisfies the following condition, if X_{m,j} ≠ {pt}, then ξ_{2}^{i,j} : C^{i} → A_{m}^{j} is injective, where Q′_{2} = ψ_{2}(P).

(3) For any blocks A_{n}^{i} ⊂ A_{n}, A_{m}^{j} ⊂ A_{m}, and for the partial maps ψ_{0}^{′i,j} and ψ_{1}^{′i,j}, we have that ψ_{0}^{′i,j}(P^{i}) := Q_{0}^{′i,j} is a projection and ψ_{1}^{′i,j} has property PE(J · rank(Q_{0}^{′i,j}), η).

Proof: Obviously, the first part of the corollary follows from Theorem 4.37. To prove the second part, we only need to perturb ψ_{0} ∈ Map(A, Q_{0}BQ_{0})_{1} to newψ_{0} such that the restriction newψ_{0}|_{D} is a homomorphism, where

D := ⨁_{i} C^{|} · P^{i} ⊕ ⨁_{i} C^{|} · (1_{A_{n}^{i}} − P^{i})

is a finite dimensional subalgebra of A_{n}. By Lemma 1.6.8, such perturbation exists if ψ_{0} is sufficiently multiplicative, which is automatically true if the set F is large enough and the number ε is small enough, using the next lemma. (Note that C := ξ_{1}(P)(⨁_{i=1}^{t_{n}} M_{[n,i]}(C[0, 1]))ξ_{1}(P) is still a direct sum of matrix algebras over C[0, 1], since all the projections in M_{•}(C[0, 1]) are trivial.)

⊓ ⊔

Lemma 4.40. Let A be a unital C^{∗}-algebra. Suppose that G ⊂ A is a finite set containing 1_{A}, and G_{1} = G × G := {gh | g ∈ G, h ∈ G}. Suppose that δ > 0, and δ′ = \frac{1}{3} \frac{1}{∥G∥}δ, where ∥G∥ = max_{g∈G}\{∥g∥\}. Suppose that B is a unital C^{∗}-algebra and p ∈ B is a projection. If a homomorphism ϕ ∈ Hom(A, B) and two maps ϕ_{1} ∈ Map(A, pBp), ϕ_{2} ∈ Map(A, (1 − p)B(1 − p)) satisfy

∥ϕ(g) − ϕ_{1}(g) ⊕ ϕ_{2}(g)∥ < δ′, ∀g ∈ G_{1},

then both ϕ_{1} and ϕ_{2} are G-δ multiplicative.

Proof: The proof is straight forward, we omit it. Theorem 4.37 and Corollary 4.39 will be used in the proof of our Main Reduction Theorem in this article. Theorem 4.35 will be used in the proof of the Uniqueness Theorem in [EGL]. The rest of this section will not be used in this paper. They are important to [EGL].

4.41. In the rest of this section, we will compare the decompositions of two different homomorphisms. Such comparison will be used in the proof of the Uniqueness Theorem in [EGL]. Let X, η, δ, ϕ, {β_{i}}i=1^{L}, and Θ(y) be as in 4.34. (Take J = 1.) Suppose that ϕ : C(X) → M_{k}(C(Y)) is as in Theorem 4.35, and ψ : C(X) → M_{k}(C(Y)) is another homomorphism with ϕ(1) = ψ(1). If

∥AffT ϕ(f) − AffT ψ(f)∥ < \frac{δ}{4}

for all f ∈ H(η, δ, X), then by Lemma 4.29, SPϕ_{y} and SPψ_{y} can be paired within \frac{η}{2} . Since SPϕ_{y} and Θ(y) can be paired within \frac{η}{2} , SPψ_{y} and Θ(y) can be paired within η. Similar to 4.34, by Theorem 4.1, there are mutually orthogonal

projections q_{1}, q_{2}, ⋯, q_{n}, q_{n+1}, ⋯, q_{L} and q_{0} = ψ(1) − ∑i=1^{L} q_{i} such that

(1) For all y ∈ Y and f ∈ F,

∥ψ(f)(y) − q_{0}ψ(f)(y)q_{0} ⊕ ∑_{i=1}^{L} f(α∘β_{i}(y))q_{i}∥ < ε.

(2) rank(q_{0}) + 2 dim(Y) ≤ rank(q_{i}). As Remark 4.22, we can choose projections p_{i} for ϕ and q_{i} for ψ to be trivial projections with rank(p_{i}) = rank(q_{i}). (Note that, in 4.34, the number L_{2} and L_{2} + L_{1}, which serve as T_{i}, i = 1, 2, ⋯, L (i.e., T_{i} = L_{2}, for 1 ≤ i ≤ L − 1, and T_{L} = L_{2} + L_{1}) in Theorem 4.1, are very larger.) Therefore, there is a unitary u ∈ M_{k}(C(Y)) such that

uq_{i}u^{∗} = p_{i}, i = 1, 2, ⋯, L.

Let ψ̃ = Adu∘ψ. Then

∥ψ̃(f)(y) − p_{0}ψ̃(f)(y)p_{0} ⊕ ∑_{i=1}^{L} f(α∘β_{i}(y))p_{i}∥ < ε

for all y ∈ Y and f ∈ F. Note that the above decomposition has the same form as that of ϕ, even with the same projections p_{i} and the part ∑i=1^{L} f(α∘β_{i}(y))p_{i}. Also, in the part ∑i=1^{L} f(α∘β_{i}(y))p_{i}, there is a map defined by point evaluations:

ϕ′(f) = ∑_{i=1}^{n} f(x_{i})p_{i},

with {x_{1}, x_{2}, ⋯, x_{n}} η-dense in X, and rank(p_{i}) ≥ rank(p_{0}) + 2 dim(Y). This means that two different homomorphisms which are close at the level of AffT can be decomposed in the same way. This result will be useful in the proof of the Uniqueness Theorem for certain spaces X with K_{1}(C(X)) a torsion group. We summarize what we obtained as the following proposition which will be used in the proof of the Uniqueness Theorem for certain spaces X with K_{1}(C(X)) a torsion group.

Proposition 4.42. Let X be a connected simplicial complex, ε > 0, and F ⊂ C(X) be a finite set. Suppose that η ∈ (0, ε) satisfies that if dist(x, x′) < 2η, then |f(x) − f(x′)| < \frac{ε}{3} for all f ∈ F. For any δ > 0, there is an integer L > 0 and a finite set H ⊂ AffT (C(X))(= C(X)) such that the following holds.

If ϕ, ψ : C(X) → M_{k}(C(Y)) are homomorphisms with properties

(a) ϕ has sdp(\frac{η}{32}, δ);

(b) rank(ϕ(1)) ≥ 2L^{2} · 2^{L}(dim X + dim Y + 1)^{3};

(c) ϕ(1) = ψ(1) and

∥AffT ϕ(h) − AffT ψ(h)∥ < \frac{δ}{4}, ∀h ∈ H,

then there are two orthogonal projections Q_{0}, Q_{1} ∈ M_{k}(C(Y)), two maps ϕ_{0}, ψ_{0} ∈ Map(C(X), Q_{0}M_{k}(C(Y))Q_{0})_{1}, a homomorphism ϕ_{1} ∈ Hom(C(X), Q_{1}M_{k}(C(Y))Q_{1})_{1}, and a unitary u ∈ M_{k}(C(Y)) such that

(1) ϕ(1) = ψ(1) = Q_{0} + Q_{1};

(2) ∥ϕ(f) − ϕ_{0}(f) ⊕ ϕ_{1}(f)∥ < ε, and ∥(Adu ∘ ψ)(f) − ψ_{0}(f) ⊕ ϕ_{1}(f)∥ < ε for all f ∈ F;

(3) ϕ_{1} factors through C[0, 1].

(4) Q_{0} = p_{0} +p_{1} +⋯+p_{n} with rank(p_{0})+2 dim(Y) ≤ rank(p_{i}) (i = 1, 2 ⋯ n), and ϕ_{0} and ψ_{0} are defined by

ϕ_{0}(f) = p_{0}ϕ(f)p_{0} + ∑_{i=1}^{n} f(x_{i})p_{i}, ∀f ∈ C(X),

ψ_{0}(f) = p_{0}(Adu ∘ ψ)(f)p_{0} + ∑_{i=1}^{n} f(x_{i})p_{i}, ∀f ∈ C(X),

where p_{0}, p_{1}, ⋯ p_{n} are mutually orthogonal projections and {x_{1}, x_{2}, ⋯ x_{n}} ⊂ X is an ε-dense subset in X. (Comparing with Theorem 4.35, the maps ϕ_{0} and ϕ_{1} in 4.35 have been put together to form the map ϕ_{0} in the above proposition.) (In [EGL], we will prove that the above ϕ_{0} and ψ_{0} are approximately unitarily equivalent to each other to within some small number (under the condition KK(ϕ) = KK(ψ)), then so also are ϕ and ψ.)

4.43. The above proposition is not strong enough to prove the Uniqueness Theorem for homomorphisms from C(S^{1}) to M_{k}(C(Y)), since K_{1}(C(S^{1})) is infinite. Before we conclude this section, we introduce a result which can be used to deal with this case (i.e, the case S^{1}). We will discuss briefly what the problem is, and how to solve the problem. Suppose that ϕ and ψ are two homomorphisms from C(S^{1}) to another C^{∗}- algebra, For ϕ and ψ to be approximately unitarily equivalent to each other, they should agree not only on AffT (C(S^{1})) and K_{∗}(C(S^{1})), but also on the determinant functions. That is, ϕ(z)ψ(z)^{∗} should have only a small variation in the determinant, where z ∈ C(S^{1}) is the standard generator. (All these things will be made precise in [EGL].) This idea has appeared in [Ell2] and [NT]. Roughly speaking, if ϕ and ψ agree (approximately) to within ε at the level of the determinant (this will also be made precise in [EGL]), then the maps p_{0}ϕp_{0} and p_{0}(Adu ∘ ψ)p_{0} from Proposition 4.42 agree only to within \frac{rank(ϕ(1))}{rank(p_{0})}ε at the level of the determinant. So for the decomposition to be useful for the proof of the uniqueness theorem, rank(p_{0}) should not be too small compared with rank(ϕ(1)). (This will be the property (2) of Theorem 4.45 below.) On the other hand, in the decompositions of ϕ and Adu ∘ ψ, we also need the homomorphism defined by point evaluations, which by Proposition 4.42 is the same for both of these decompositions, to be large in order to absorb the parts p_{0}ϕp_{0} and p_{0}(Adu ∘ ψ)p_{0}. This will be the property (3) of Theorem 4.45. Therefore, rank(p_{0}) should not be too large either. To do that, besides the property sdp(\frac{η}{32}, δ), we also need sdp property for an extra pair (\frac{η̃}{32}, δ̃), where η̃ depends on δ. For those readers who are familiar with [Ell2] and [NT], we encourage them to compare the sdp property for the two pairs (\frac{η}{32}, δ) and (\frac{η̃}{32}, δ̃), with the conditions of Theorem 4 of [Ell2], and Lemma 2.3 and Theorem 2.4 of [NT], in the following way. In Theorem 4 of [Ell2] (see page 100 of [Ell2]), roughly speaking, the sentence on lines 16–20 corresponds to our property sdp(\frac{η̃}{32}, δ̃), and the sentence on lines 21–22 corresponds to our property sdp(\frac{η}{32}, δ). That is, \frac{1}{m} corresponds to our \frac{η}{32} (or \frac{η}{16} in some sense), \frac{3}{n} corresponds to our δ, \frac{1}{n} corresponds to our \frac{η̃}{32}, and δ corresponds to our δ̃. Similarly, in Lemma 2.3 of [NT], condition (1) corresponds to our sdp(\frac{η}{32}, δ) and condition (2) corresponds to our sdp(\frac{η̃}{32}, δ̃). Also in Theorem 2.4 of [NT], condition (2) corresponds to our sdp(\frac{η}{32}, δ) and condition (3) corresponds to our sdp(\frac{η̃}{32}, δ̃). Such a construction will be given in 4.44 below. In 4.44, we will first describe the condition that ϕ should satisfy. Then we will carry out the construction in three steps. In Step 1, we will follow the procedure in 4.34, to decompose ϕ into p_{0}ϕp_{0} ⊕ϕ_{1}, corresponding to the property sdp(\frac{η̃}{32}, δ̃) (not sdp(\frac{η}{32}, δ)). (Here, the map ϕ_{1} is ϕ_{1} ⊕ ϕ_{2} in the notation of 4.34 or 4.35.) In Step 2, we will take a part p′ϕ_{1}p′ out of ϕ_{1} and add it to p_{0}ϕp_{0} to obtain P_{0}ϕP_{0}, where P_{0} = p_{0} + p′. The rest of ϕ_{1} will be defined to be newϕ_{1}. In this way, we can get the projection P_{0} with suitable size (neither too small nor too large). The size depends on δ, which explains why η̃ depends on δ. In Step 3. we will prove that newϕ_{1} can be decomposed again in such a way that the point evaluation part of its decomposition is sufficiently large that it can be used to control P_{0}ϕP_{0}, in the proof of the uniqueness theorem in [EGL]. (See the property (3) of Theorem 4.45.) The property sdp(\frac{η}{32}, δ) is used in this step.

4.44. Let F ⊂ C(X) be a finite set, ε > 0 and ε_{1} > 0. Suppose that the positive number η < \frac{ε_{1}}{4} satisfies the condition that, if dist(x, x′) < 2η, then

∥f(x) − f(x′)∥ < \frac{ε}{3}.

For any δ > 0, consider the pair (η, δ) as in 4.33. Let N, n be as in 4.33. Instead of choosing L = n(N + 2), we choose

L ≥ max\{n(N + 2), \frac{8}{δ}, \frac{4}{ε}, \frac{4}{ε_{1}} \}.

Consider ε̃ = \frac{1}{8L} < min(ε, ε_{1}). Let positive number η̃ < \frac{η}{4} satisfy that, if dist(x, x′) < 2η̃, then

∥f(x) − f(x′)∥ < \frac{ε̃}{3}.

Let δ̃ > 0 be any number. Then for the pair (η̃, δ̃), there exists an integer L̃ playing the role of L as in Lemma 4.33. We can assume L̃ > L. Let

Λ = 6L̃^{2} · 2_{L̃}(dim X + M + 1)^{3},

where M is a positive integer. Now let Y be a simplicial complex with dim Y ≤ M, and ϕ : C(X) → P M_{k}(C(Y))P be a unital homomorphism satisfying the following two conditions:

(a) ϕ has both sdp(\frac{η}{32}, δ) and sdp(\frac{η̃}{32}, δ̃);

(b) rank(P) ≥ Λ. We will construct a decomposition for ϕ. Step 1. By the discussion in 4.34 corresponding to sdp(\frac{η̃}{32}, δ̃), there is a set

Θ(y) = {α∘β_{1}(y)^{∼L_{2}}, α∘β_{2}(y)^{∼L_{2}}, ⋯, α∘β_{_{L̃}−1}(y)^{∼L_{2}}, α∘β_{L̃}(y)^{∼L_{2}+L_{1}}} ,

where

L_{2} = int (\frac{rank(P)}{L̃}) ≥ int (\frac{Λ}{L̃}) ,

such that SPϕ_{y} and Θ(y) can be paired within \frac{η̃}{2}. As in 4.34, there are mutually orthogonal projections p_{0} and P_{1} = ∑_{L̃} ; i=1 p_{i} and a homomorphism ϕ_{1} : C(X) → P_{1}M_{k}(C(Y))P_{1}, such that

(1) ∥ϕ(f) − p_{0}ϕ(f)p_{0} ⊕ ϕ_{1}(f)∥ ≤ ε̃ ≤ \frac{1}{8L},

(2) rank(p_{0}) ≤ L̃ · 2_{L̃}(dim X + M + 1)^{3} ≤ int (\frac{Λ}{6L̃}), where ϕ_{1} is defined by

ϕ_{1}(f)(y) = ∑_{i=1}^{_{L̃}} f(α∘β_{i}(y))p_{i}

with rank(p_{i}) ≥ L_{2} − 2_{L̃}(dim X + M + 1)^{3}. Step 2. We will take a part p′ϕ_{1}p′ out from ϕ_{1} and add it into p_{0}ϕp_{0}, such that the projection P_{0} = p_{0} + p′ has rank about \frac{rank(P)}{L}, which is neither too large nor too small. (Here we use L not L̃.) There exists a projection p′ satisfying the following two conditions.

(c) p′ = ∑_{L̃} ; i=1 p′_{i}, with p′_{i} < p_{i}, i = 1, 2, ⋯, L̃.

(d) rank(p′) = int (\frac{rank(P)}{L}) (here we use L, not L̃), where L was chosen in the beginning of this subsection. We can make the above (d) hold for the following reason. First,

rank(∑_{i=1}^{L̃} p_{i}) ≥ rank(P) − int (\frac{Λ}{6L̃}) > int (\frac{rank(P)}{L}) + L̃ dim(Y).

So one can choose non negative integers k_{1}, k_{2}, ⋯, k_{L̃} such that ∑L̃ ; i=1 k_{i} =

int (\frac{rank(P)}{L}) and that k_{i} ≤ rank(p_{i}) − dim(Y). Therefore, by [Hu], we can choose trivial projections p′_{i} < p_{i} with rank(p′_{i}) = k_{i}. Define

P_{0} = p_{0} ⊕ p′  and  newP_{1} = P_{1} ⊖ p′.

Note that p′ is a sub-projection of P_{1} = ∑_{L̃} ; i=1 p_{i}. Define newϕ_{1} : C(X) → newP_{1}M_{k}(C(Y))newP_{1} by

(newϕ_{1}(f))(y) = ∑_{i=1}^{_{L̃}} f(α∘β_{i}(y))(p_{i} ⊖ p′_{i}).

newP_{1} and newϕ_{1} are still denoted by P_{1} and ϕ_{1}, respectively. Evidently, the following are true.

(1′)  ∥ϕ(f) − P_{0}ϕ(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4L}.

(2′)  \frac{rank(P)}{L} ≤ rank(P_{0}) ≤ 2 · int (\frac{rank(P)}{L}) .

(Notice that, to get the above decomposition, one only needs the condition that SPϕ_{y} and Θ(y) can be paired within η̃ (see the way η is chosen in 4.4 for Theorem 4.1 and the way η̃ is chosen above). On the other hand, SPϕ_{y} and Θ(y) can be paired within \frac{η̃}{2} in our case. So if ψ satisfies the condition that SPψ_{y} and SPϕ_{y} can be paired within \frac{η̃}{2}, then the above decomposition also holds for ψ, as discussed in 4.41. In particular, for a certain unitary u, Adu ∘ ψ can have same form of decomposition as ϕ does— same projection P_{0} and even exactly the same part of the above ϕ_{1}. This will be used in 4.46 and Proposition 4.47.) Step 3. Now, we can decompose ϕ_{1} again to obtain a large part of the homomorphism defined by point evaluations, which will be used to absorb the part of P_{0}ϕP_{0}, in the proof of the uniqueness theorem in [EGL]. For the compact metric space X, and η > 0 (now we use η not η̃), there exists a finite η-dense subset {x_{1}, x_{2}, ⋯, x_{m}} such that dist(x_{i}, x_{j}) ≥ η, if i ≠ j.

(Such set could be chosen to be a maximum set of finite many points which have mutual distance at least η. Then the η-density of the set follows from the maximality.) We will prove the following claim. Claim: There are mutually orthogonal projections q_{1}, q_{2}, ⋯, q_{m} < P_{1} with rank(q_{i}) > rank(P_{0}) + dim(Y), such that

∥ϕ_{1}(f) − (P_{1} − ∑_{i=1}^{m} q_{i})ϕ_{1}(f)(P_{1} − ∑_{i=1}^{m} q_{i}) ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}∥ < ε

for all f ∈ F. Proof of the claim: First, we know that the set (SPϕ_{1})_{y} is obtained by deleting rank(P_{0}) points (counting multiplicity) from the set Θ(y). Also SPϕ_{y} and Θ(y) can be paired within \frac{η̃}{2}. Recall that,

Θ(y) = {α∘β_{1}(y)^{∼L_{2}}, α∘β_{2}(y)^{∼L_{2}}, ⋯, α∘β_{_{L̃}−1}(y)^{∼L_{2}}, α∘β_{L̃}(y)^{∼L_{2}+L_{1}}}

is the set corresponding to ϕ and the pair (η̃, δ̃) in 4.33. And recall that L_{2} = int (\frac{rank(P)}{L̃}). From (a), ϕ has the property sdp(\frac{η}{32}, δ). So Θ(y) has the

property sdp(\frac{η}{32} + \frac{η̃}{2}, δ). But (SPϕ_{1})_{y} is obtained by deleting

rank(P_{0}) (≤ 2 · int (\frac{rank(P)}{L}) ≤ \frac{δ}{4}rank(P))

points from Θ(y). (Note that \frac{1}{L} < \frac{δ}{8}.) Therefore, in the (\frac{η}{32} + \frac{η̃}{2})-ball of any point in X, (SPϕ_{1})_{y} contains at least

δ · rank(P) − \frac{δ}{4}rank(P) = \frac{3δ}{4} rank(P)

points (counting multiplicity). That is, ϕ_{1} has the property sdp(\frac{η}{32} + \frac{η̃}{2}, \frac{3δ}{4} ). Therefore ϕ_{1} has the property sdp(\frac{η}{4} , \frac{3δ}{4} ), since η̃ < \frac{η}{4} . Set U_{i} = B_{\frac{η}{2}} (x_{i}), i = 1, 2, ⋯, m. Then U_{i}, i = 1, 2, ⋯, m are mutually disjoint open sets, since dist(x_{i}, x_{j}) ≥ η, if i ≠ j. By the property sdp(\frac{η}{4} , \frac{3δ}{4} ) of ϕ_{1}, for any y ∈ Y,

#(SP(ϕ_{1})_{y} ∩ U_{i}) ≥ \frac{3δ}{4} rank(P) > \frac{2}{L}rank(P) + 3 dim Y > rank(P_{0}) + 3 dim(Y).

The claim follows from the following proposition: Proposition. Let X be a simplicial complex, and F ⊂ C(X) a finite subset. Let ε > 0 and η > 0 be such that if dist(x, x′) < 2η, then |f(x) − f(x′)| < \frac{ε}{3} for any f ∈ F. Suppose that U_{1}, U_{2}, ⋯, U_{m} are disjoint open neighborhoods of points x_{1}, x_{2}, ⋯, x_{m} ∈ X, respectively, such that U_{i} ⊂ B_{η}(x_{i}) for all 1 ≤ i ≤ m.

Suppose that ϕ : C(X) → P M_{•}(C(Y))P is a unital homomorphism, where Y is a simplicial complex, such that

#(SPϕ_{y} ∩ U_{i}) ≥ k_{i}, for 1 ≤ i ≤ m and for all y ∈ Y.

Then there are mutually orthogonal projections q_{1}, q_{2}, ⋯, q_{m} ∈ P M_{•}(C(Y))P with rank(q_{i}) ≥ k_{i} − dim(Y) such that

∥ϕ(f) − p_{0}ϕ(f)p_{0} ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}∥ < ε, for all f ∈ F,

where p_{0} = P − ∑ q_{i}. This is Proposition 1.5.7 of this paper (see 1.5.4—1.5.6 for the proof). Since the expert reader may skip §1.5, we point out that the above result was essentially proved in [EG2, Theorem 2.21]. So we obtain the projections q_{i} with

rank(q_{i}) ≥ min_{y}(#(SP(ϕ_{1})_{y} ∩ U_{i})) − dim(Y) ≥ rank(p_{0}) + 2 dim(Y).

Summarizing the above, we obtain the following theorem.

Theorem 4.45. Let F ⊂ C(X) be a finite set, ε > 0, ε_{1} > 0, and let M be a positive integer (in the application in [EGL], we will let M = 3). Let the positive number η < \frac{ε_{1}}{4} satisfy that, if dist(x, x′) < 2η, then

∥f(x) − f(x′)∥ < \frac{ε}{3} for all f ∈ F.

Let δ > 0 be any positive number. There is an integer L > max\{\frac{8}{δ}, \frac{4}{ε}, \frac{4}{ε_{1}} \} satisfying the following condition. The rest of the theorem describes this condition. Suppose that η̃ > 0 satisfies that, if dist(x, x′) < 2η̃, then

∥f(x) − f(x′)∥ < \frac{1}{24L} for all f ∈ F.

For any δ̃ > 0, there is a positive integer Λ such that if a unital homomorphism ϕ : C(X) → P M_{k}(C(Y))P (with dim Y ≤ M) satisfies the following conditions

(a) ϕ has the properties sdp(\frac{η}{32}, δ) and sdp(\frac{η̃}{32}, δ̃);

(b) rank(P) ≥ Λ, then there are projections P_{0}, P_{1} ∈ P M_{k}(C(Y))P (with P_{0} + P_{1} = P) and a homomorphism ϕ_{1} : C(X) → P_{1}M_{k}(C(Y))P_{1} such that

(1) ∥ϕ(f) − P_{0}ϕ(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4L} for all f ∈ F;

(2) rank(P_{0}) ≥ \frac{rank(P)}{L};

(3) There are mutually orthogonal projections q_{1}, q_{2},⋯, q_{m} ∈ P_{1}M_{k}(C(Y))P_{1} and an η-dense finite subset {x_{1}, x_{2}, ⋯, x_{m}} ⊂ X with the following properties.

(i) rank(q_{i}) > rank(P_{0}) + 2 dim(Y), i = 1, 2, ⋯, m; (ii) ∥ϕ_{1}(f) − (P_{1} − ∑_{i=1}^{m} q_{i})ϕ_{1}(f)(P_{1} − ∑_{i=1}^{m} q_{i}) ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}∥ < ε for all f ∈ F.

4.46. Let η̃ and δ̃ be as in 4.44 (or 4.45), and H(η̃, δ̃, X) ⊂ C(X) the subset defined in 4.27. Suppose that ϕ : C(X) → P M_{k}(C(Y))P satisfies the conditions (a) and (b) in Theorem 4.45. And suppose that ψ : C(X) → P M_{k}(C(Y))P is another homomorphism satisfying

∥AffT ϕ(h) − AffT ψ(h)∥ < \frac{δ̃}{4},

for all h ∈ H(η̃, δ̃, X). Similar to 4.41, there is a unitary u ∈ P M_{k}(C(Y))P such that

∥Adu∘ψ(f) − P_{0}Adu∘ψ(f)P_{0} ⊕ ϕ_{1}∥ < \frac{1}{4L}, ∀f ∈ F

where P_{0} and ϕ_{1} are exactly the same as those for ϕ in Theorem 4.45. (See the end of step 2 of 4.44.) So we have the following proposition.

Proposition 4.47. Let F ⊂ C(X) be a finite set, ε > 0, ε_{1} > 0, and let M be a positive integer (in the application in [EGL], we will let M = 3). Let the positive number η < \frac{ε_{1}}{4} satisfy that, if dist(x, x′) < 2η, then

∥f(x) − f(x′)∥ < \frac{ε}{3} for all f ∈ F.

Let δ > 0 be any positive number. There is an integer L > max\{\frac{8}{δ}, \frac{4}{ε}, \frac{4}{ε_{1}} \} satisfying the following condition. The rest of the proposition describes this condition. Suppose that η̃ > 0 satisfies that, if dist(x, x′) < 2η̃, then

∥f(x) − f(x′)∥ < \frac{1}{24L} for all f ∈ F.

For any δ̃ > 0, there is a positive integer Λ and a finite set H ⊂ AffT (C(X))(= C(X)) such that if unital homomorphisms ϕ, ψ : C(X) → P M_{k}(C(Y))P (with dim Y ≤ M) satisfy the following conditions:

(a) ϕ has the properties sdp(\frac{η}{32}, δ) and sdp(\frac{η̃}{32}, δ̃);

(b) rank(P) ≥ Λ;

(c) ∥AffT ϕ(h) − AffT ψ(h)∥ < \frac{δ̃}{4}, ∀h ∈ H, then there are projections P_{0}, P_{1} ∈ P M_{k}(C(Y))P (with P_{0} + P_{1} = P), a homomorphism ϕ_{1} : C(X) → P_{1}M_{k}(C(Y))P_{1} factoring through C[0, 1], and a unitary u ∈ P M_{k}(C(Y))P such that

(1) ∥ϕ(f) − P_{0}ϕ(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4L} and ∥(Adu ∘ ψ)(f) − P_{0}(Adu ∘ ψ)(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4L} for all f ∈ F; (2) rank(P_{0}) ≥ \frac{rank(P)}{L};

(3) There are mutually orthogonal projections q_{1}, q_{2},⋯, q_{m} ∈ P_{1}M_{k}(C(Y))P_{1} and an η-dense finite subset {x_{1}, x_{2}, ⋯, x_{m}} ⊂ X with the following properties.

(i) rank(q_{i}) > rank(P_{0}) + 2 dim(Y); (ii) ∥ϕ_{1}(f) − (P_{1} − ∑_{i=1}^{m} q_{i})ϕ_{1}(f)(P_{1} − ∑_{i=1}^{m} q_{i}) ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}∥ < ε for all f ∈ F. In order to be consistent in notation with the application in [EGL], let us rewrite the above proposition in the following form.

Proposition 4.47’. For any finite set F ⊂ C(X), ε > 0, ε_{1} > 0, there is a number η > 0 with the property described below. For any δ > 0, there are an integer K > \frac{4}{ε} and a a number η̃ > 0 satisfying the following condition. For any δ̃ > 0, there is a positive integer L and a finite set H ⊂ AffT (C(X))(H can be chosen to be H(η̃, δ̃, X) in 4.27) such that if two unital homomorphisms ϕ, ψ : C(X) → P M_{k}(C(Y))P (with dim Y ≤ 3) satisfy the following conditions:

(a) ϕ has the properties sdp(\frac{η}{32}, δ) and sdp(\frac{η̃}{32}, δ̃);

(b) rank(P) ≥ L;

(c) ∥AffT ϕ(h) − AffT ψ(h)∥ < \frac{δ̃}{4}, ∀h ∈ H, then there are projections P_{0}, P_{1} ∈ P M_{k}(C(Y))P (with P_{0} + P_{1} = P), a homomorphism ϕ_{1} : C(X) → P_{1}M_{k}(C(Y))P_{1} factoring through C[0, 1], and a unitary u ∈ P M_{k}(C(Y))P such that

(1) ∥ϕ(f) − P_{0}ϕ(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4K} and ∥(Adu ∘ ψ)(f) − P_{0}(Adu ∘ ψ)(f)P_{0} ⊕ ϕ_{1}(f)∥ < \frac{1}{4K} for all f ∈ F;

(2) rank(P_{0}) ≥ \frac{rank(P)}{K};

(3) There are mutually orthogonal projections q_{1}, q_{2},⋯, q_{m} ∈ P_{1}M_{k}(C(Y))P_{1} and an \frac{ε_{1}}{4} -dense finite subset \{x_{1}, x_{2}, ⋯, x_{m}\} ⊂ X with the following properties.

(i) rank(q_{i}) > rank(P_{0}) + 2 dim(Y); (ii) ∥ϕ_{1}(f) − (P_{1} − ∑_{i=1}^{m} q_{i})ϕ_{1}(f)(P_{1} − ∑_{i=1}^{m} q_{i}) ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}∥ < ε for all f ∈ F. (Notice that in the above statement, we change the notation L and Λ to K and L respectively. Also, in condition (3), we change η-density to \frac{ε_{1}}{4} -density.)

4.48. Proposition 4.47’ will be used in the proof of the Uniqueness Theorem in [EGL]. Namely, we will prove that, under certain conditions about KK(ϕ) and KK(ψ) and the determinants of ϕ(z) and ψ(z) (see (4) of Theorem 2.4 of [NT] ), where z ∈ C(S^{1}) is the standard generator,

P_{0}ϕ(f)P_{0} ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}, f ∈ F

is approximately unitarily equivalent to

P_{0}Adu∘ψ(f)P_{0} ⊕ ∑_{i=1}^{m} f(x_{i})q_{i}, f ∈ F.

Therefore, {ϕ(f), f ∈ F } is approximately unitarily equivalent to {ψ(f), f ∈ F }. In [EGL], we need both of the following conditions:

rank(P_{0}) ≥ \frac{rank(P)}{L}  and  [q_{i}] > [P_{0}] in K_{0}(C(Y)).

In comparison with Theorem 2.4 of [NT], in the Uniqueness Theorem in [EGL], we also have a condition similar to (4) of Theorem 2.4 of [NT]. But this condition will be useful only when it is combined with the condition (2) above (see [EGL] for details).

5 Almost Multiplicative Maps

In this section, we study almost multiplicative maps

ϕ ∈ Map (M_{l}(C(X)), M_{l_{1}}(C(Y))) ,

where X = T_{II,k}, T_{III,k}, or S^{2}, and Y is a simplicial complex of dimension at most M with M a fixed number. In this section, all the simplicial complexes are assumed to have dimension at most M.

5.1. Suppose that B_{1}, B_{2}, ⋯, B_{n}, ⋯ are unital C^{∗}-algebras. Let B = ⨁_{n=1}^{+∞} B_{n}. Then the multiplier algebra M(B) of B is ∏_{n=1}^{+∞} B_{n}. The Six Term Exact Sequence associated to

0 ⟶ B ⟶ M(B) ⟶ M(B)/B ⟶ 0

breaks into two exact sequences

0 ⟶ K_{0}(B) ⟶ K_{0}(M(B)) ⟶ K_{0}(M(B)/B) ⟶ 0 and
0 ⟶ K_{1}(B) ⟶ K_{1}(M(B)) ⟶ K_{1}(M(B))/B ⟶ 0.

since each projection (or unitary) in M_{n}(M(B)/B) can be lifted to a projection (or a unitary) in M_{n}(M(B)). Furthermore,

K_{0}(B) = ⨁_{n=1}^{+∞} K_{0}(B_{n})  and  K_{1}(B) = ⨁_{n=1}^{+∞} K_{1}(B_{n}).

But in general, it is not true that

K_{0}(M(B)) = ∏_{n=1}^{+∞} K_{0}(B_{n})  or  K_{1}(M(B)) = ∏_{n=1}^{+∞} K_{1}(B_{n}).

In fact, K_{0}(M(B)) is a subgroup of ∏_{n=1}^{+∞} K_{0}(B_{n}). But K_{1}(M(B)) is more complicated. In the first part of this section, we will calculate the K-theory of M(B) (and of M(B)/B) for the case

B_{n} = M_{k_{n}}(C(X_{n})),

where X_{n} are simplicial complexes of dimension at most M. For convenience, we always suppose that the spaces X_{n} are connected.

5.2. Consider S^{1} = {z; |z| = 1} ⊂ C^{|} . Let

F : (S^{1}\{−1}) × [0, 1] ⟶ S^{1}\{−1}

be defined by

F (e^{iθ}, t) = e^{itθ}, −π < θ < π.

Then |t − t′| < ε implies

|F (x, t) − F (x, t′)| < πε.

This fact implies the following. If u and v are unitaries such that ∥u − v∥ < 1, then there is a path of unitaries u_{t} with u_{0} = u, u_{1} = v such that |t − t′| < ε implies ∥u_{t} − u_{t′}∥ < πε. Let SU(n)(⊂ U(n)) denote the collection of n × n unitaries with determinant 1. Let SU_{n}(X) denote the collection of continuous functions from X to SU(n). Note SU_{n}(X) ⊂ U_{n}(X) ⊂ M_{n}(C(X)). From the proof of Theorem 3.3 (and Lemma 3.1) of [Phi2] (in particular (∗ ∗ ∗) in Step 4 of 3.3 of [Phi2]), one can prove the following useful fact.

Lemma 5.3. ([Phi2]) For each positive integer M, there is an M′ > 0 satisfying the following condition. For any connected finite CW-complex X of dimension at most M, and u, v ∈ SU_{n}(X), if u and v can be connected to each other in U_{n}(X), then there is a path u_{t} ∈ SU_{n}(X) such that

1. u_{0} = u, u_{1} = v and

2. |t − t′| < ε implies ∥u_{t} − u_{t′}∥ < M′ · ε. (Note that M′ does not depend on n, the size of the unitaries.)

5.4. Let B_{n} = M_{k_{n}}(C(X_{n})), dim(X_{n}) ≤ M. Let B = ⨁_{n=1}^{+∞} B_{n}. Then we can describe K_{0}(M(B)) as below. Let (K_{0}(B_{n}), K_{0}(B_{n})^{+}, 1_{B_{n}}) be the scaled ordered K-group of B_{n} (see 1.2 of [EG2]). Let Π_{b}K_{0}(B_{n}) be the subgroup of ∏_{n=1}^{+∞} K_{0}(B_{n}) consisting of elements

(x_{1}, x_{2}, ⋯, x_{n}, ⋯) ∈ ∏_{n=1}^{+∞} K_{0}(B_{n})

with the property that there is a positive integer L such that

−L[1_{B_{n}}] < x_{n} < L[1_{B_{n}}] ∈ K_{0}(B_{n}) for all n.

Lemma 5.5. K_{0} (∏_{n=1}^{+∞} B_{n}) = Π_{b}K_{0}(B_{n}).

Proof: Any element in K_{0} (∏_{n=1}^{+∞} B_{n}) is of the form [p] − [q], where p, q ∈

M_{L} (∏_{n=1}^{+∞} B_{n}) are projections. Let

p = (p_{1}, p_{2}, ⋯, p_{n}, ⋯), q = (q_{1}, q_{2}, ⋯, q_{n}, ⋯) ∈ M_{L} (∏_{n=1}^{+∞} B_{n}) .

Then [p] − [q] ∈ K_{0} (∏_{n=1}^{+∞} B_{n}) corresponds to the element

([p_{1}] − [q_{1}], [p_{2}] − [q_{2}], ⋯, [p_{n}] − [q_{n}], ⋯) ∈ Π_{b}K_{0}(B_{n}).

We will prove that this correspondence is bijective. Surjectivity: Let

([p_{1}] − [q_{1}], [p_{2}] − [q_{2}], ⋯, [p_{n}] − [q_{n}], ⋯) ∈ Π_{b}K_{0}(B_{n}).

Then there is an L > M such that

−L[1_{B_{n}}] < [p_{n}] − [q_{n}] < L[1_{B_{n}}], ∀n.

Therefore,

−L · k_{n} ≤ rank(p_{n}) − rank(q_{n}) < L · k_{n}, ∀n.

It is well known that (see [Hu]) any vector bundle of dimension M + T over an M dimensional space has a T dimensional trivial sub-bundle. Thus one can replace p_{n} by p′_{n}, q_{n} by q′_{n}, with properties

[p′_{n}] < 2L[1_{B_{n}}], [q′_{n}] < 2L[1_{B_{n}}] and
[p′_{n}] − [q′_{n}] = [p_{n}] − [q_{n}] in K_{0}(B_{n}).

([p′_{1}] − [q′_{1}], [p′_{2}] − [q′_{2}], ⋯) is in the image of the correspondence, since every element [p′_{n}] < 2L[1_{B_{n}}] can be realized by a projection in M_{4L}(B_{n}) (recall that L > M). Injectivity. Let p = (p_{1}, p_{2}, ⋯, p_{n}, ⋯) and q = (q_{1}, q_{2}, ⋯, q_{n}, ⋯) be projections in M_{L} (∏_{n=1}^{+∞} B_{n}). Suppose that for each n, [p_{n}] = [q_{n}] ∈ K_{0}(B_{n}). We have to prove that [(p_{1}, p_{2}, ⋯, p_{n}, ⋯)] = [(q_{1}, q_{2}, ⋯, q_{n}, ⋯)] ∈ K_{0} (∏_{n=1}^{+∞} B_{n}) . Without loss of generality, assume that L > M. Let 1_{n} ∈ M_{L}(B_{n}) be the unit. By [Hu], for each n, the projection p_{n} ⊕ 1_{n} is unitary equivalent to q_{n} ⊕ 1_{n}. That is, there is a unitary u_{n} ∈ M_{2L}(B_{n}) such that q_{n} ⊕ 1_{n} = u_{n}(p_{n} ⊕ 1_{n})u_{n}^{∗}. Hence the unitary u = (u_{1}, u_{2}, ⋯, u_{n}, ⋯) ∈ M_{2L}(∏_{n=1}^{+∞} B_{n}) satisfies q ⊕ 1 = u(p ⊕ 1)u^{∗}. It follows that [q] = [p].

⊓ ⊔

5.6. Let X be a finite CW complex. Then K_{1}(C(X)) = K^{1}(X) is defined to be the collection of homotopy equivalence classes of continuous maps from X to U(∞), denoted by [X, U(∞)], (or from X to U(n), denoted by [X, U(n)], for n large enough). Consider the fibration

SU(n) ⟶ U(n) ⟶^{b} S^{1},

where S^{1} ⊂ C^{|} is the unit circle, b is defined by sending a unitary to its determinant, and SU(n) is the special unitary group consisting the unitaries of

determinant 1. The fibration has a splitting S^{1} ⟶^{b^{−1}} U(n), defined by

S^{1} ∋ z ⟼^{b^{−1}} ( \begin{matrix}z \\ & 1 \\ & & . & . & . \\ & & & & & 1 \end{matrix} ) ∈ U(n).

One can identify U(n) with SU(n) × S^{1} by U(n) ∋ u ↦ ((b^{−1} ∘ b(u))^{∗}u, b(u)) ∈ SU(n) × S^{1}. Therefore, [X, U(n)] = [X, SU(n)] ⊕ [X, S^{1}] as a group. We use notation SK_{1}(C(X)) or SK^{1}(X) to denote [X, SU(n)], n large enough, and π^{1}(X) to denote [X, S^{1}]. Then

K_{1}(C(X)) = SK_{1}(C(X)) ⊕ π^{1}(X).

(The splitting is not a natural splitting.)

5.7. Let {X_{n}} be a sequence of connected finite CW complexes of dimension at most M. Let B_{n} = M_{k_{n}}(C(X_{n})). Define a map

τ : K_{1} (∏_{n=1}^{+∞} B_{n}) ⟶ ∏_{n=1}^{+∞} K_{1}B_{n} by

τ[(u_{1}, u_{2}, ⋯, u_{n}, ⋯)] = ([u_{1}], [u_{2}], ⋯, [u_{n}], ⋯),

where (u_{1}, u_{2}, ⋯, u_{n}, ⋯) is a unitary in M_{L} (∏_{n=1}^{+∞} B_{n}). If L ≥ M, then any element in K_{1}(B_{n}) can be realized by a unitary in M_{L}(B_{n}). Based on this fact, we know that τ is surjective. We will prove that

0 ⟶ Kerτ ⟶ K_{1} (∏_{n=1}^{+∞} B_{n}) ⟶ ∏_{n=1}^{+∞} K_{1}B_{n} ⟶ 0

is a splitting exact sequence. A splitting

τ̃ : ∏_{n=1}^{+∞} K_{1}B_{n} ⟶ K_{1} (∏_{n=1}^{+∞} B_{n})

will be defined such that τ ∘ τ̃ = id on ∏_{n=1}^{+∞} K_{1}B_{n}. By 5.6,

K_{1}B_{n} = SK_{1}B_{n} ⊕ π^{1}(X_{n}).

Hence we define τ̃ on ∏_{n=1}^{+∞} SK_{1}B_{n} and ∏_{n=1}^{+∞} π^{1}(X_{n}) separately. If x ∈ ∏_{n=1}^{+∞} SK_{1}B_{n} is represented by a sequence of unitaries

u_{1} ∈ M_{L}(B_{1}), u_{2} ∈ M_{L}(B_{2}), ⋯, u_{n} ∈ M_{L}(B_{n}), ⋯,

each with determinant 1, then define

τ̃(x) = [(u_{1}, u_{2}, ⋯, u_{n}, ⋯)] ∈ K_{1} (∏_{n=1}^{+∞} B_{n}) .

To see that τ̃ is well defined, let v_{1}, v_{2}, ⋯, v_{n}, ⋯ be another sequence with determinant 1 and

[u_{n}] = [v_{n}] in K_{1}(B_{n}).

Without loss of generality, we assume that L > M. By Lemma 5.3, for each n, there is a unitary path u_{n} such that u_{n}(0) = u_{n}, u_{n}(1) = v_{n}, and ∥u_{n}(t) − u_{n}(t′)∥ < M′ ·|t−t′|, ∀t ∈ [0, 1], where M′ is a constant which does not depend on n. Obviously,

(u_{1}(t), u_{2}(t), ⋯, u_{n}(t), ⋯) ∈ (M_{L}(∏_{n=1}^{+∞} B_{n})) ⊗ C([0, 1]).

Hence

[(u_{1}, u_{2}, ⋯, u_{n}, ⋯)] = [(v_{1}, v_{2}, ⋯, v_{n}, ⋯)].

That is, the above map is well defined. (Warning: It is not enough to prove that each u_{n} can be connected to v_{n}, since a sequence of paths, each connecting u_{n} and v_{n} (n = 1, 2, ⋯), only defines an element in M_{L}(∏_{n=1}^{+∞}(B_{n} ⊗ C[0, 1])), but

(∏_{n=1}^{+∞} B_{n}) ⊗ C[0, 1] \begin{matrix}⊂ \\ ≠ \end{matrix} ∏_{n=1}^{+∞}(B_{n} ⊗ C[0, 1]). )

The following claim is a well known folklore result in topology. Since we can not find a precise reference, we present a proof here. Claim: For any connected simplicial complex X, the cohomotopy group π^{1}(X) is a finitely generated free abelian group. Proof of the claim. Let X^{(1)} be the 1-skeleton of X. Then X^{(1)} is homotopy equivalent to a finite wedge of S^{1}. Evidently, π^{1}(X^{(1)}) is a finitely generated free abelian group. (In comparison with the above cohomotopy group, we point out that the fundamental group π_{1}(X^{(1)}) of a finite wedge X^{(1)} of S^{1} is a free group (not a free abelian group).)

On the other hand, we can prove that

i^{∗} : π^{1}(X) → π^{1}(X^{(1)}),

induced by the inclusion i : X^{(1)} → X, is an injective map as below. Once this is done, the claim follows from the result in group theory that any subgroup of a free abelian group is still a free abelian group. Let us prove the injectivity of i^{∗}. Suppose that f, g : X → S^{1} are two maps satisfying that

i^{∗}([f]) = i^{∗}([g]),

where [f], [g] ∈ π^{1}(X) are elements represented by f and g, respectively. Then f|_{X^{(1)}} is homotopic to g|_{X^{(1)}}. Let F : X^{(1)} × [0, 1] → S^{1} be a homotopy path connecting f|_{X^{(1)}} and g|_{X^{(1)}}. That is

F |_{X^{(1)}×{0}} = f|_{X^{(1)}} and F |_{X^{(1)}×{1}} = g|_{X^{(1)}}.

We are going to extend the homotopy F to a homotopy on the entire space X × [0, 1]. The construction is done by induction. Suppose that F has been extended to a homotopy (let us still denote it by F) F : X^{(n)} × [0, 1] → S^{1} between f|_{X^{(n)}} and g|_{X^{(n)}} on the n-skeleton (where n ≥ 1) of X. I.e., F |_{X^{(n)}×{0}} = f|_{X^{(n)}} and F |_{X^{(n)}×{1}} = g|_{X^{(n)}}. We need to prove that it can be extended to a homotopy on the (n + 1)-skeleton. Let Δ be any (n + 1)-simplex. Then ∂Δ ⊂ X^{(n)}. Let G : ∂Δ × [0, 1] ∪ Δ × {0} ∪ Δ × {1} → S^{1} be defined by

G(x) = \{ \begin{matrix}F (x) & if x ∈ ∂Δ × [0, 1] \\ f(x) & if x ∈ Δ × \{0\} \\ g(x) & if x ∈ Δ × \{1\} . \end{matrix}

Then G(x) is a continuous map from ∂(Δ × [0, 1]) to S^{1}. Since π_{n+1}(S^{1}) = 0 and ∂(Δ × [0, 1]) = S^{n+1}, G can be extended to a map G : Δ × [0, 1] → S^{1}. Define F on each simplex Δ to be this G. Then F is the desired extension. This ends the proof of the claim. Let us go back to the construction of τ̃ on ∏_{n=1}^{+∞} π^{1}(X_{n}). Let x_{1}^{0} ∈ X_{1}, x_{2}^{0} ∈ X_{2}, ⋯, x_{n}^{0} ∈ X_{n}, ⋯, be chosen as the base points of the spaces. Let

θ_{n,1}, θ_{n,2}, ⋯, θ_{n,t_{n}} : X_{n} ⟶ S^{1}

be the functions representing the generators

[θ_{n,1}], [θ_{n,2}], ⋯, [θ_{n,t_{n}}] ∈ π^{1}(X_{n}).

Suppose that

θ_{n,j}(x_{n}^{0}) = 1 ∈ S^{1} ⊂ C^{|} , j = 1, 2, ⋯, t_{n}.

For any element (x_{1}, x_{2}, ⋯, x_{n}, ⋯) ∈ ∏_{n=1}^{+∞} π^{1}(X_{n}), define τ̃(x) as below. Let u_{n} ∈ B_{n} be defined by

u_{n}(y) = ( \begin{matrix}θ_{n,1}(y)^{m_{1}} · θ_{n,2}(y)^{m_{2}} ⋯ θ_{n,t_{n}}(y)^{m_{tn}} \\ & 1 \\ & & . & . & . \\ & & & & & 1 \end{matrix} )_{k_{n}×k_{n}}∈ M_{k_{n}}(C^{|} )

for each y ∈ X_{n}, where m_{1}, m_{2}, ⋯, m_{t_{n}} are integers with

(∗) x_{n} = m_{1}[θ_{n,1}] + m_{2}[θ_{n,2}] + ⋯ + m_{t_{n}}[θ_{n,t_{n}}] ∈ π^{1}(X_{n}).

Define

τ̃(x) = [(u_{1}, u_{2}, ⋯, u_{n}, ⋯)] ∈ K_{1} (∏_{n=1}^{+∞} B_{n}) .

Since each π^{1}(X_{n}) is a free abelian group, the expression (∗) for x_{n} is unique. It is easy to check that τ̃ on ∏_{n=1}^{+∞} π^{1}(X_{n}) is a well defined group homomorphism. It is straight forward to check that

τ ∘ τ̃ = id : ∏_{n=1}^{+∞} π^{1}(X_{n}) ⟶ ∏_{n=1}^{+∞} π^{1}(X_{n})

And that

τ ∘ τ̃ = id : ∏_{n=1}^{+∞} SK_{1}(B_{n}) ⟶ ∏_{n=1}^{+∞} SK_{1}(B_{n}).

That is,

τ ∘ τ̃ = id : ∏_{n=1}^{+∞} K_{1}(B_{n}) ⟶ ∏_{n=1}^{+∞} K_{1}(B_{n}).

The splitting τ̃ : ∏_{n=1}^{+∞} K_{1}(B_{n}) ⟶ K_{1} (∏_{n=1}^{+∞} B_{n}) of the exact sequence

0 ⟶ Ker(τ) ⟶ K_{1} (∏_{n=1}^{+∞} B_{n}) ⟶^{τ} ∏_{n=1}^{+∞} K_{1}B_{n} ⟶ 0

gives an isomorphism

K_{1} (∏_{n=1}^{+∞} B_{n}) = ∏_{n=1}^{+∞} K_{1}B_{n} ⊕ Ker(τ).

5.8. In order to identify Ker(τ), suppose that

u = [(u_{1}, u_{2}, ⋯, u_{n}, ⋯)] ∈ K_{1} (∏_{n=1}^{+∞} B_{n})

satisfies that

τ(u) = 0 ∈ ∏_{n=1}^{+∞} K_{1}B_{n}.

Note that any unitary matrix v ∈ M_{•}(C^{|} ) can be connected to 1 ∈ M_{•}(C^{|} ), by a path v(t) satisfying that, if |t − t′| < ε, then

∥v(t) − v(t′)∥ < 2πε.

Based on this fact, we have

[(u_{1}, u_{2}, ⋯)] = [(u_{1}^{∗}(x_{1}^{0})u_{1}, u_{2}^{∗}(x_{2}^{0})u_{2}, ⋯)] ∈ K_{1} (∏_{n=1}^{+∞} B_{n}) .

Therefore, without loss of generality, we assume that

u_{n}(x_{n}^{0}) = 1 ∈ M_{L}(B_{n}),

where x_{n}^{0} ∈ X_{n} are the base points. Since τ(u) = 0, if we assume L ≥ M, then each u_{n} can be connected to 1 ∈ M_{L}(B_{n}). This implies that the map

determinant(u_{n}) : X_{n} ⟶ S^{1}

is homotopy trivial. Therefore, this map can be lifted to a unique map

det(u_{n}) : X_{n} ⟶ IR

such that det(u_{n})(x_{n}^{0}) = 0 ∈ IR and

exp(2πi det(u_{n})) = determinant(u_{n}).

Let ∐ X_{n} be the disjoint union of X_{n} and Map (∐ X_{n}, IR)_{0} the set of all continuous maps f : ∐ X_{n} → IR with f(x_{n}^{0}) = 0 for all x_{n}^{0}. Let Map_{b} (∐ X_{n}, IR)_{0} be the set of those maps with bounded images.

Define a map d : Ker(τ) → Map(∐_{n=1}^{+∞} X_{n},IR)_{0} ; Map_{b}(∐_{n=1}^{+∞} X_{n},IR)_{0} by

d(u) = [(\frac{det(u_{1})}{k_{1}}, \frac{det(u_{2})}{k_{2}}, ⋯, \frac{det(u_{n})}{k_{n}}, ⋯)].

We will prove that d is a well defined isomorphism. Suppose that u can be represented by another unitary

(v_{1}, v_{2}, ⋯, v_{n}, ⋯) ∈ M_{L} (∏_{n=1}^{+∞} B_{n})

with v(x_{n}^{0}) = 1 ∈ M_{L}(B_{n}). Then for the unit of a certain matrix algebra over ∏_{n=1}^{+∞} B_{n}

1_{_{L}_{1}} ∈ M_{_{L}_{1}} (∏_{n=1}^{+∞} B_{n}) ,

we have that, the element

(u_{1} ⊕ 1, u_{2} ⊕ 1, ⋯, u_{n} ⊕ 1, ⋯) ∈ M_{L+L_{1}} (∏_{n=1}^{+∞} B_{n})

can be connected to the element

(v_{1} ⊕ 1, v_{2} ⊕ 1, ⋯, v_{n} ⊕ 1, ⋯) ∈ M_{L+L_{1}} (∏_{n=1}^{+∞} B_{n})

by a unitary path

(u_{1}(t), u_{2}(t), ⋯, u_{n}(t), ⋯) ∈ (M_{L+L_{1}} (∏_{n=1}^{+∞} B_{n})) ⊗ C[0, 1].

We need to prove that

(\frac{det(u_{1}) − det(v_{1})}{k_{1}}, \frac{det(u_{2}) − det(v_{2})}{k_{2}}, ⋯, \frac{det(u_{n}) − det(v_{n})}{k_{n}}, ⋯)

has a uniformly bounded image in IR. This follows from the following fact. If two unitaries w_{1}, w_{2} ∈ M_{(L+L_{1})k_{n}}(C^{|} ) satisfying |w_{1} − w_{2}| < ε < \frac{1}{4}, then

|determinant(w_{1}^{∗}w_{2}) − 1| < π(L + L_{1})k_{n}ε.

Now, we have to prove that d is an isomorphism. Obviously, d is surjective. In fact, for any function

(f_{1}, f_{2}, ⋯, f_{n}, ⋯) ∈ Map (∐_{n=1}^{+∞} IR)_{0} ,

let u ∈ K_{1} (∏_{n=1}^{+∞} B_{n}) be the element represented by

(exp(2πif_{1}), exp(2πif_{2}), ⋯, exp(2πif_{n}), ⋯) ∈ ∏_{n=1}^{+∞} B_{n} = ∏_{n=1}^{+∞} M_{k_{n}}(C(X_{n})).

Then d(u) = [(f_{1}, f_{2}, ⋯, f_{n}, ⋯)]. Finally, we have to prove that d is injective. Suppose that u ∈ Ker(τ) is represented by

(u_{1}, u_{2}, ⋯, u_{n}, ⋯) ∈ M_{L} (∏_{n=1}^{+∞} B_{n})

satisfying

(\frac{det(u_{1})}{k_{1}}, \frac{det(u_{2})}{k_{2}}, ⋯, \frac{det(u_{n})}{k_{n}}, ⋯) ∈ Map_{b} (∐ X_{n}, IR)_{0} .

Let f_{n} = \frac{det(u_{n})}{k_{n}} : X_{n} → IR and

v_{n} = exp\frac{2πif_{n}}{L} ∈ M_{Lk_{n}}(C(X_{n})).

Then v_{n}^{∗}u_{n} ∈ SU_{Lk_{n}}(X_{n}), i.e., it has determinant 1 every where. Since (f_{1}, f_{2}, ⋯, f_{n}, ⋯) is of uniformly bounded image, we know that

(u_{1}, u_{2}, ⋯, u_{n}, ⋯)  and  (v_{1}^{∗}u_{1}, v_{2}^{∗}u_{2}, ⋯ v_{n}^{∗}u_{n}, ⋯)

can be connected by a continuous path in

(M_{L} (∏_{n=1}^{+∞} B_{n})) ⊗ C[0, 1].

Therefore, u = [(v_{1}^{∗}u_{1}, v_{2}^{∗}u_{2}, ⋯ v_{n}^{∗}u_{n}, ⋯)]. The latter is zero by Lemma 5.3 and the fact that u ∈ Ker(τ). Summarizing the above, we obtain

Lemma 5.9. K_{1} (∏_{n=1}^{+∞} B_{n}) = ∏_{n=1}^{+∞} K_{1}B_{n} ⊕ Map(∐_{n=1}^{+∞} X_{n},IR)_{0} ; Map_{b}(∐_{n=1}^{+∞} X_{n},IR)_{0}

= ∏_{n=1}^{+∞} SK_{1}B_{n} ⊕ ∏_{n=1}^{+∞} π^{1}(X_{n}) ⊕ Map (∐_{n=1}^{+∞} X_{n}, IR)_{0} ; Map_{b} (∐_{n=1}^{+∞} X_{n}, IR)_{0} .

Corollary 5.10.

K_{1} (∏_{n=1}^{+∞} B_{n}/ ⨁_{n=1}^{+∞} B_{n}) = (∏_{n=1}^{+∞} K_{1}B_{n}/ ⨁_{n=1}^{+∞} K_{1}B_{n}) ⊕ Map (∐_{n=1}^{+∞} X_{n}, IR)_{0} ; Map_{b} (∐_{n=1}^{+∞} X_{n}, IR)_{0} .

5.11. From [Sch], for any C^{∗}-algebra A in the bootstrap class and any C^{∗}- algebra B (not necessarily separable), there is a splitting short exact sequence

0 ⟶ K_{∗}(A) ⊗ K_{∗}(B) ⟶ K_{∗}(A ⊗ B) ⟶ T or(K_{∗}(A), K_{∗}(B)) ⟶ 0.

Let A = C_{0}(W_{k}), where W_{k} = T_{II,k} as in the introduction. W_{k} is used for T_{II,k} only when involving mod k K-theory K_{∗}(B, ZZ/k). From the definition

K_{∗}(B, ZZ/k) := K_{∗}(A ⊗ B),

one has

0 ⟶ K_{0}(B) ⊗ ZZ/k ⟶ K_{0}(B, ZZ/k) ⟶ T or(ZZ/k, K_{1}(B)) ⟶ 0

and

0 ⟶ K_{1}(B) ⊗ ZZ/k ⟶ K_{1}(B, ZZ/k) ⟶ T or(ZZ/k, K_{0}(B)) ⟶ 0.

Since G ⊗ ZZ/k can be identified with the cokernel of

G ⟶^{×k} G,

and Tor(ZZ/k, G) can be identified with the kernel of

G ⟶^{×k} G,

one has the following well known exact sequences

K_{0}(B) ⟶^{×k} K_{0}(B) ⟶ K_{0}(B, ZZ/k) ⟶ K_{1}(B) ⟶^{×k} K_{1}(B)

and

K_{1}(B) ⟶^{×k} K_{1}(B) ⟶ K_{1}(B, ZZ/k) ⟶ K_{0}(B) ⟶^{×k} K_{0}(B).

5.12. Let {X_{n}}_{n=1}^{+∞}, B_{n} = M_{k_{n}}(C(X_{n})) be as above, and B = ⨁_{n=1}^{+∞} B_{n}, M(B) = ∏_{n=1}^{+∞} B_{n}, Q(B) = M(B)/B. From Lemma 5.5 and Corollary 5.10, we have

K_{0}(Q(B)) = Π_{b}K_{0}(B_{n})/ ⊕ K_{0}(B_{n}) and

K_{1}(Q(B)) = \{\frac{∏_{n=1}^{+∞} K_{1}(B_{n})}{⨁_{n=1}^{+∞} K_{1}(B_{n})}\} ⨁ \begin{cases}Map & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \\ Map & b & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \end{cases} \} .

It is easy to see that the map

\begin{cases}Map & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \\ Map & b & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \end{cases} \} ⟶^{×k} \begin{cases}Map & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \\ Map & b & (∐_{n=1}^{+∞} X_{n}, IR) & 0 \end{cases} \}

is an isomorphism. Any torsion element x_{n} ∈ K_{0}(B_{n}) can be realized as a formal difference of two projections p, q ∈ M_{∞}(B_{n}) of the same rank. (The rank of a projection makes sense since X_{n} are connected. Also for any element x ∈ K_{0}(B_{n}) represented by [p] − [q], we define rank(x) = rank(p) − rank(q), which is always a (possibly negative) integer.) By [Hu], if a projection p ∈ M_{•}(C(X_{n})) has rank larger than dim(X_{n}) + r, then p has a trivial sub projection of rank r. Therefore, any torsion element x_{n} ∈ K_{0}(B_{n}) can be realized as a formal difference of two projections p, q ∈ M_{L}(B_{n}) if L > dim(X_{n}). Based on this fact, one can directly compute that

Kernel (Π_{b}K_{0}(B_{n}) ⟶^{×k} Π_{b}K_{0}(B_{n})) = Kernel (∏_{n=1}^{+∞} K_{0}(B_{n}) ⟶^{×k} ∏_{n=1}^{+∞} K_{0}(B_{n})) .

Fixed a positive integer k. let x = (x_{1}, x_{2}, ⋯, x_{n}, ⋯) ∈ ∏_{n=1}^{+∞} K_{0}(B_{n}). For each element x_{n} ∈ K_{0}(B_{n}), one can write rank(x_{n}) = k · M · l_{n} + r_{n}, where l_{n} is a (possibly negative) integer, M is the maximum of {dim(X_{n})}_{n}, and 0 < k ·M ≤ r_{n} < 2k ·M. Let p_{n} be the trivial rank one projection in B_{n}. Then x_{n} can be written as k · M · l_{n}[p_{n}] + [q_{n}], where q_{n} is a projection of rank r_{n}. Therefore, x can can be written as x = x′ + x′′, where x′ ∈ k(∏_{n=1}^{+∞} K_{0}(B_{n})) and x′′ ∈ Π_{b}K_{0}(B_{n}). As a consequence, one can compute that

Cokernel (Π_{b}K_{0}(B_{n}) ⟶^{×k} Π_{b}K_{0}(B_{n}))

= Cokernel (∏_{n=1}^{+∞} K_{0}(B_{n}) ⟶^{×k} ∏_{n=1}^{+∞} K_{0}(B_{n})) .

Combined with 5.11, yields

K_{0}(Q(B), ZZ/k) = ∏_{n=1}^{+∞} K_{0}(B_{n}, ZZ/k)/ ⨁_{n=1}^{+∞} K_{0}(B_{n}, ZZ/k),

and

K_{1}(Q(B), ZZ/k) = ∏_{n=1}^{+∞} K_{1}(B_{n}, ZZ/k)/ ⨁_{n=1}^{+∞} K_{1}(B_{n}, ZZ/k).

5.13. Following [DG], denote

\underline{K}(A) = K_{∗}(A) ⊕ ⨁_{n=2}^{+∞} K_{∗}(A, ZZ/n).

For any finite CW complex X and two KK-elements α, β ∈ KK(C(X), A), from [DL] (also see [DG]), we know that α = β if and only if

α_{∗} = β_{∗} : \underline{K}(C(X)) ⟶ \underline{K}(A).

We will discuss the special cases of X = {pt}, [0, 1], T_{II,k}, T_{III,k} and S^{2}, where T_{II,k}, T_{III,k} are defined in the Introduction. (See §4 of [EG2] for details.) (The case X = {pt} or [0, 1] is similar to the case X = S^{2}, so we will not discuss the spaces {pt} and [0, 1] separately.) From [DL], there is an isomorphism

KK(C(X), B) ⟶ Hom_{Λ}(\underline{K}(C(X)), \underline{K}(B)), where Hom_{Λ}(\underline{K}(C(X)), \underline{K}(B)) is the set of systems of group homomorphisms which is compatible with all the Bockstein Operations (see [DL] for details). For any fixed finite CW complex X, an element α ∈ KK(C(X), B) is determined by the system of maps

α_{n}^{∗} : K_{∗}(C(X), ZZ/n) ⟶ K_{∗}(B, ZZ/n), n = 0, 2, 3, ⋯

which are induced by α. In fact α would be determined by a few maps from the above list—all the other maps in the system \{α_{n}^{∗}\}_{n=0}^{+∞} : \underline{K}(C(X)) → \underline{K}(B) would be completely determined by these few maps via the Bockstein Operations. We will choose those few maps for the cases X = {pt}, [0, 1], S^{2}, T_{II,k}, or T_{III,k}.

1. X = S^{2}. Then

\underline{K}(C(S^{2})) ⟶ \underline{K}(B)

is completely determined by

K_{0}(C(S^{2})) ⟶ K_{0}(B)

via the Bockstein Operation

K_{0}(C(S^{2})) ⟶ K_{0}(C(S^{2}), ZZ/k)
↓  ↓
K_{0}(B) ⟶ K_{0}(B, ZZ/k)

since the top horizontal map is surjective. (Note that K_{1}(C(S^{2})) and K_{1}(C(S^{2}), ZZ/k) are trivial groups.) Therefore,

KK(C(S^{2}), B) ≅ Hom(K_{0}(C(S^{2})), K_{0}(B)).

(This is also a well known consequence of the Universal Coefficient Theorem.) (The case X = {pt} or [0, 1] is similar to the above case.)

2. X = T_{II,k}. Let rC(T_{II,k}) ≅ C^{|} and let C_{0}(T_{II,k}) be the ideal of C(T_{II,k}) consisting of the continuous functions vanishing at the base point. (See 1.6 of [EG2] and 1.1.7 for the notations.) Consider the splitting exact sequence

0 ⟶ K_{0}(C_{0}(T_{II,k})) ⟶ K_{0}(C(T_{II,k})) ⟶ K_{0}(rC(T_{II,k})) ⟶ 0.

Each KK-element α ∈ KK(C(T_{II,k}), B) induces two group homomorphisms

α_{0}^{0} : K_{0}(rC(T_{II,k})) (= ZZ) ⟶ K_{0}(B) and
α_{k}^{1} : K_{1}(C(T_{II,k}), ZZ/k) (= ZZ/k) ⟶ K_{1}(B, ZZ/k).

This induces a map

KK(C(T_{II,k}),B)⟶Hom(K_{0}(rC(T_{II,k})),K_{0}(B)) ⨁ Hom(K_{1}(C(T_{II,k}),ZZ/k),K_{1}(B,ZZ/k))

= Hom(ZZ,K_{0}(B)) ⨁ Hom(ZZ/k,K_{1}(B,ZZ/k)).

It can be verified that any two homomorphisms

α_{0}^{0} : K_{0}(rC(T_{II,k})) (= ZZ) ⟶ K_{0}(B) and
α_{k}^{1} : K_{1}(C(T_{II,k}), ZZ/k) (= ZZ/k) ⟶ K_{1}(B, ZZ/k)

induces a unique system of homomorphisms in Hom_{Λ}(\underline{K}(C(T_{II,k})), \underline{K}(B)). Therefore, the above map is an isomorphism. Another way to see it, is as follows. Note that

K_{1}(C(T_{II,k}), ZZ/k) = K_{0}(C_{0}(T_{II,k})) ⊂ K_{0}(C(T_{II,k})).

Considering

K_{1}(B) ⟶^{×k} K_{1}(B) ⟶ K_{1}(B, ZZ/k) ⟶ K_{0}(B) ⟶^{×k} K_{0}(B),

we obtain

Hom(K_{1}(C(T_{II,k}), ZZ/k), K_{1}(B, ZZ/k))

≅ Hom(K_{0}(C_{0}(T_{II,k})), K_{0}(B)) ⊕ Ext(K_{0}(C_{0}(T_{II,k})), K_{1}(B)).

Then from the Universal Coefficient Theorem,

KK(C(T_{II,k}), B) ≅ Hom(K_{0}(C(T_{II,k})), K_{0}(B)) ⊕ Ext(K_{0}(C(T_{II,k})), K_{1}(B)) ≅ Hom(K_{0}(rC(T_{II,k})), K_{0}(B)) ⊕ Hom(K_{0}(C_{0}(T_{II,k})), K_{0}(B))

⊕Ext(K_{0}(C_{0}(T_{II,k})), K_{1}(B)).

(Note that K_{1}(C(T_{II,k})) = 0.) Hence one can see again, the map mentioned above is an isomorphism.

3. X = T_{III,k}. Also, let rC(T_{III,k}) = C^{|} and let C_{0}(T_{III,k}) be the ideal consisting of functions vanishing at the base point. Notice that

K_{0}(C(T_{III,k})) = ZZ  and  K_{0}(C_{0}(T_{III,k}), ZZ/k) = ZZ/k.

By the splitting exact sequence

0 →K_{0}(C_{0}(T_{III,k}), ZZ/k) → K_{0}(C(T_{III,k}), ZZ/k) → K_{0}(rC(T_{III,k}), ZZ/k) → 0

we know that each α ∈ KK(C(T_{III,k}), B) induces an element

α_{k}^{0} : K_{0}(C_{0}(T_{III,k}), ZZ/k) ⟶ K_{0}(B, ZZ/k).

It can be proved that

KK(C(T_{III,k}), B)

≅ Hom(K_{0}(C(T_{III,k})), K_{0}(B)) ⨁ Hom(K_{0}(C_{0}(T_{III,k}), ZZ/k), K_{0}(B, ZZ/k))

= Hom(ZZ, K_{0}(B)) ⨁ Hom(ZZ/k, K_{0}(B, ZZ/k)), as what we did for T_{II,k}. (Notice that the map K_{1}(C(T_{III,k})) → K_{1}(B) is completely determined by the map K_{0}(C_{0}(T_{III,k}), ZZ/k) → K_{0}(B, ZZ/k).) Summarizing the above, we have the following. For any two elements α, β ∈ KK(C(X), B), α = β if and only if

(1) α_{0}^{0} = β_{0}^{0} : K_{0}(C(X)) ⟶ K_{0}(B), when X = S^{2};

(2) α_{0}^{0} = β_{0}^{0} : K_{0}(rC(X)) ⟶ K_{0}(B) and α_{k}^{1} = β_{k}^{1} : K_{1}(C(X), ZZ/k) ⟶ K_{1}(B, ZZ/k), when X = T_{II,k};

(3) α_{0}^{0} = β_{0}^{0} : K_{0}(C(X)) ⟶ K_{0}(B) and α_{k}^{0} = β_{k}^{0} : K_{0}(C_{0}(X), ZZ/k) ⟶ K_{0}(B, ZZ/k), when X = T_{III,k}. Therefore, we have the following lemma.

Lemma 5.14. Let A = P M_{l}(C(X))P, and X one of {pt}, [0, 1], T_{II,k}, T_{III,k}, or S^{2}. Let α, β ∈ KK(A, B), where B is a C^{∗}-algebra. Then α = β if and only if the following hold:

1. When X = {pt}, [0, 1] or S^{2},

α_{∗} = β_{∗} : K_{0}(A) ⟶ K_{0}(B);

2. When X = T_{II,k},

α_{∗} = β_{∗} : K_{0}(A) ⟶ K_{0}(B) and
α_{∗} = β_{∗} : K_{1}(A, ZZ/k) ⟶ K_{1}(B, ZZ/k);

3. When X = T_{III,k},

α_{∗} = β_{∗} : K_{0}(A) ⟶ K_{0}(B) and
α_{∗} = β_{∗} : K_{0}(A, ZZ/k) ⟶ K_{0}(B, ZZ/k).

Combined with Theorem 6.1 of [DG], yields the following lemma.

Lemma 5.15. Let A = P M_{l}(C(X))P, and X, one of {pt}, [0, 1], T_{II,k}, T_{III,k} or S^{2}, and let B be any C^{∗}-algebra. Let ϕ, ψ ∈ Hom(A, B). Suppose that the following statements hold.

1. When X = {pt}, [0, 1] or S^{2},

[ϕ]_{∗} = [ψ]_{∗} : K_{0}(A) ⟶ K_{0}(B);

2. When X = T_{II,k},

[ϕ]_{∗} = [ψ]_{∗} : K_{0}(A) ⟶ K_{0}(B) and
[ϕ]_{∗} = [ψ]_{∗} : K_{1}(A, ZZ/k) ⟶ K_{1}(B, ZZ/k);

3. When X = T_{III,k},

[ϕ]_{∗} = [ψ]_{∗} : K_{0}(A) ⟶ K_{0}(B) and

[ϕ]_{∗} = [ψ]_{∗} : K_{0}(A, ZZ/k) ⟶ K_{0}(B, ZZ/k).

It follows that, for any finite set F ⊂ A and any number ε > 0, there exist n ∈ IN, μ ∈ Hom(A, M_{n}(B)) with finite dimensional image and a unitary u ∈ M_{n+1}(B) such that

∥u(ϕ(a) ⊕ μ(a))u^{∗} − ψ(a) ⊕ μ(a)∥ < ε

for all a ∈ F.

5.16. Fix A = P M_{l}(C(X))P, X = {pt}, [0, 1], T_{II,k}, T_{III,k} or S^{2}. Then A is stably isomorphic to C(X). By 5.14, an element α ∈ KK(A, B) is completely determined by

α_{0}^{0} : K_{0}(A) → K_{0}(B),
α_{k}^{0} : K_{0}(A, ZZ/k) → K_{0}(B, ZZ/k), and
α_{k}^{1} : K_{1}(A, ZZ/k) → K_{1}(B, ZZ/k).

Note that, for any C^{∗}-algebra A,

K_{0}(A ⊗ C(W_{k} × S^{1})) ≅ K_{0}(A) ⊕ K_{1}(A) ⊕ K_{0}(A, ZZ/k) ⊕ K_{1}(A, ZZ/k).

Each projection p ∈ M_{∞}(A ⊗ C(W_{k} × S^{1})) defines an element

[p] ∈ K_{0}(A) ⊕ K_{1}(A) ⊕ K_{0}(A, ZZ/k) ⊕ K_{1}(A, ZZ/k) ⊂ \underline{K}(A).

This defines a map from the set of projections in ⋃_{k=2}^{∞} M_{∞}(A ⊗ C(W_{k} × S^{1})) to \underline{K}(A). For any finite set P ⊂ ⋃_{k=2}^{∞} M_{∞}(A ⊗ C(W_{k} × S^{1})) of projections, denoted by P\underline{K}(A) the finite subset of \underline{K}(A) consisting of elements coming from the projections p ∈ P, that is

P\underline{K}(A) = \{[p] ∈ \underline{K}(A)| p ∈ P\}.

In particular, if A = P M_{l}(C(X))P, X = T_{II,k}, or T_{III,k}, then we can choose a finite set of projections P_{A} ⊂ M_{•}(A ⊗ C(W_{k} × S^{1})) such that the set {[p] ∈ K_{0}(A) ⊕ K_{1}(A) ⊕ K_{0}(A, ZZ/k) ⊕ K_{1}(A, ZZ/k) | p ∈ P_{A}\} = P_{A}\underline{K}(A) generates K_{0}(A) ⊕ K_{1}(A) ⊕ K_{0}(A, ZZ/k) ⊕ K_{1}(A, ZZ/k) ⊂ \underline{K}(A). For X = \{pt\}, [0, 1] or S^{2}, choose P_{A} ⊂ M_{•}(A) such that {[p] ∈ K_{0}(A) | p ∈ P_{A}} generates K_{0}(A) ⊂ \underline{K}(A). We will use P to denote P_{A} if there is no danger of confusion.

5.17. Let A = P M_{l}(C(X))P, X = {pt}, [0, 1], T_{II,k}, T_{III,k} or S^{2}, and P ⊂ M_{•}(A ⊗ C(W_{k} × S^{1})) or P ⊂ M_{•}(A) be as in 5.16. There are a finite subset G(P) ⊂ A and a number δ(P) > 0 such that if B is any C^{∗}-algebra and ϕ ∈ Map(A, B) is G(P) − δ(P) multiplicative, then

∥((ϕ ⊗ id)(p))^{2} − (ϕ ⊗ id)(p)∥ < \frac{1}{4}, ∀p ∈ P,

where id is the identity map on M_{•}(C(W_{k} × S^{1})) or on M_{•}(C^{|} ). Hence for any p ∈ P, there is a projection q ∈ M_{•}(B ⊗ C(W_{k} × S^{1})) (or q ∈ M_{•}(B) ) such that

∥(ϕ ⊗ id)(p) − q∥ < \frac{1}{2}.

So q defines an element in \underline{K}(B). (If q′ is another projection satisfying the same condition, then ∥q − q′∥ < 1, hence q′ is unitarily equivalent to q.) Therefore, if ϕ is G(P) − δ(P) multiplicative, then it induces a map

ϕ_{∗} : P\underline{K}(A) → \underline{K}(B).

Note that such G(P) and δ(P) could be defined for any finite set P ⊂ M_{∞}(A)∪ M_{∞}(A ⊗ C(S^{1})) ∪ ⋃_{k=2}^{∞} M_{∞}(A ⊗ C(W_{k} × S^{1})) of projections.

Theorem 5.18. Let X be one of the spaces {pt}, [0, 1], T_{II,k}, T_{III,k} or S^{2}. Let A = P M_{l}(C(X))P and P be as in 5.16. For any finite set F ⊂ A, any positive number ε > 0, and any positive integer M, there are a finite set G ⊂ A (G ⊃ G(P) large enough ), a positive number δ > 0 (δ < δ(P) small enough), and a positive integer L (large enough) such that the following statement is true. If ϕ, ψ ∈ Map(A, B) are G-δ multiplicative and

ϕ_{∗} = ψ_{∗} : P\underline{K}(A) ⟶ \underline{K}(B),

where B = QM_{•}(C(Y))Q with dim(Y) ≤ M, then there is a homomorphism ν ∈ Hom(A, M_{L}(B)), with finite dimensional image, and there is a unitary u ∈ M_{L+1}(B) such that

∥u(ϕ ⊕ ν)(a)u^{∗} − (ψ ⊕ ν)(a)∥ < ε, ∀f ∈ F.

Proof: We first prove the theorem for the case B = M_{•}(C(Y)). Then we apply Lemma 1.3.6 to reduce the general case to this special case. We prove the theorem by contradiction. Let G(P) ⊂ G_{1} ⊂ G_{2} ⊂ ⋯ ⊂ G_{n} ⊂ ⋯ be a sequence of finite subsets with

\overline{∪G_{n}} = unit ball of A.

Let δ(P) > δ_{1} > δ_{2} > ⋯ > δ_{n} > ⋯ be a sequence of positive numbers with δ_{n} → 0. Let L_{1} < L_{2} < ⋯ < L_{n} < ⋯ be a sequence of positive integers with L_{n} → +∞. Suppose that the theorem does not hold for (G_{n}, δ_{n}, L_{n}). That is, there exist a C^{∗}-algebra B_{n} = M_{k_{n}}(C(Y_{n})) and two G_{n}-δ_{n} multiplicative maps

ϕ_{n}, ψ_{n} : A ⟶ B_{n}

with (ϕ_{n})_{∗} = (ψ_{n})_{∗} : P\underline{K}(A) ⟶ \underline{K}(B_{n}) and

inf_{ν,u} sup_{a∈F} ∥u(ϕ_{n} ⊕ ν)(a)u^{∗} − (ψ_{n} ⊕ ν)(a)∥ ≥ ε, (∗)

where ν runs over all subsets of Hom(A, M_{L_{n}}(B_{n})) consisting of those homomorphisms with finite dimensional images, and u runs over U(M_{L_{n}+1}(B_{n})). The above {ϕ_{n}}_{n=1}^{+∞}, {ψ_{n}}_{n=1}^{+∞} induce two homomorphisms

ϕ̃, ψ̃ : A ⟶ ∏_{n=1}^{+∞} B_{n}/ ⨁_{n=1}^{+∞} B_{n} = Q(B).

We will prove that KK(ϕ) = KK(ψ).

1. X = {pt}, [0, 1] or S^{2}. By Lemma 5.14, KK(ϕ̃) is completely determined by

([ϕ̃]_{∗})_{0}^{0} : K_{0}(A) ⟶ K_{0}(Q(B)).

From 5.12,

K_{0}(Q(B)) = Π_{b}K_{0}(B_{n})/ ⨁_{n=1}^{+∞} K_{0}(B_{n}).

That is, the above ([ϕ̃]_{∗})_{0}^{0} is completely determined by the component ([ϕ_{n}]_{∗})_{0}^{0}. From the condition that

[ϕ_{n}]_{∗} = [ψ_{n}]_{∗} : P\underline{K}(A) ⟶ \underline{K}(B_{n})

and the condition that the group generated by P\underline{K}(A) is K_{0}(A), we know that KK(ϕ) = KK(ψ).

2. X = T_{II,k}. By Lemma 5.14, KK(ϕ̃) is completely determined by

([ϕ̃]_{∗})_{0}^{0} : K_{0}(A) ⟶ K_{0}(Q(B)) and
([ϕ̃]_{∗})_{k}^{1} : K_{1}(A, ZZ/k) ⟶ K_{1}(Q(B), ZZ/k).

Furthermore, by 5.12,

K_{1}(Q(B), ZZ/k) = ∏_{n=1}^{+∞} K_{1}(B_{n}, ZZ/k)/ ⨁_{n=1}^{+∞} K_{1}(B_{n}, ZZ/k).

Again, ([ϕ̃]_{∗})_{0}^{0} and ([ϕ̃]_{∗})_{k}^{1} are completely determined by the components corresponding to [ϕ_{n}]_{∗}. And from [ϕ_{n}]_{∗} = [ψ_{n}]_{∗} on P\underline{K}(A), we obtain KK(ϕ̃) = KK(ψ̃). (Note that, we also use the fact that P\underline{K}(A) generates a subgroup of \underline{K}(A) containing K_{0}(A) and K_{1}(A, ZZ/k).) (The subgroup of \underline{K}(A) generated by P\underline{K}(A) also contains K_{1}(A), though we do not use this fact.)

3. X = T_{III,k}. It can be proved that KK(ϕ̃) = KK(ψ̃) as above. Note that K_{0}(B_{n}, ZZ/k) = ∏_{n=1}^{+∞} K_{0}(B_{n}, ZZ/k)/ ⨁_{n=1}^{+∞} K_{0}(B_{n}, ZZ/k), by 5.12.

By Lemma 5.15, there are a positive integer L and a homomorphism ν̃ : A → M_{L}(∏_{n=1}^{+∞} B_{n}/ ⨁_{n=1}^{+∞} B_{n}) with finite dimensional image, and a unitary ũ ∈ M_{L+1}(∏_{n=1}^{+∞} B_{n}/ ⨁_{n=1}^{+∞} B_{n}), such that

∥ũ(ϕ̃ ⊕ ν̃)(a)ũ^{∗} − (ψ̃ ⊕ ν̃)(a)∥ < \frac{ε}{2}

for all a ∈ F. Since ν̃ has finite dimensional image, one can find a sequence of homomorphisms

ν_{n} : A ⟶ M_{L}(B_{n})

of finite dimensional images such that {ν_{n}}_{n=1}^{+∞} induces ν̃. One can also lift ũ to a sequence of unitaries u_{n} ∈ M_{L+1}(B_{n}). Then if n is large enough, we have

∥u_{n}(ϕ_{n} ⊕ ν_{n})(a)u_{n}^{∗} − (ψ_{n} ⊕ ν_{n})(a)∥ < ε

for all a ∈ F. This contradicts with (∗) if one choose n to satisfy L_{n} ≥ L. Now we apply Lemma 1.3.6 to prove the general case. Let G, δ and L_{1} (in place of L) be as above for the case of full matrix algebras over C(Y) with dim(Y) ≤ M. Choose L = (2M + 2)L_{1} − 1. We will verify that G, δ and L satisfies the condition of the theorem even for B = QM_{•}(C(Y))Q—cutting down of full matrix algebras by projections, as follows. Let n = rank(Q) + dim(Y) and m = 2M + 1. Then by Lemma 1.3.6, QM_{•}(C(Y))Q can be identified as a corner subalgebra of M_{n}(C(Y)), and M_{n}(C(Y)) can be identified as corner subalgebra of M_{m}(QM_{•}(C(Y))Q). If ϕ, ψ ∈ Map_{G−δ}(A, B) satisfy the condition in the theorem, then regarding B as a corner subalgebra of M_{n}(C(Y)), we can regard ϕ, ψ as elements in Map_{G−δ}(A, M_{n}(C(Y))) which still satisfy the condition. Hence from the above special case of the theorem, there are ν : A → M_{_{L}_{1}}(M_{n}(C(Y))) and a unitary u_{1} ∈ M_{L_{1}+1}(M_{n}(C(Y))) such that

∥u_{1}(ϕ ⊕ ν)(a)u_{1}^{∗} − (ψ ⊕ ν)(a)∥ < ε, ∀a ∈ F.

Also, M_{n}(C(Y)) can be regarded as a corner subalgebra of M_{m}(QM_{•}(C(Y))Q), so ϕ ⊕ ν and ψ ⊕ ν can be regarded as maps from A to M_{L_{1}+1}(M_{m}(QM_{•}(C(Y))Q)) = M_{L+1}(QM_{•}(C(Y))Q). Therefore, there is a unitary u ∈ M_{L+1}(B)

∥u(ϕ ⊕ ν)(a)u^{∗} − (ψ ⊕ ν)(a)∥ < ε, ∀a ∈ F.
⊓ ⊔

Remark 5.19. The theorem is not true for X = S^{1}, even if we assume that both ϕ and ψ are homomorphisms. A counterexample is given below . Let ϕ_{n}, ψ_{n} : C(S^{1}) → C[0, 1] be defined by

ϕ_{n}(f)(t) = f(e^{2πint})  and  ψ_{n}(f)(t) = f(1).

Then KK(ϕ_{n}) = KK(ψ_{n}). Let F = \{z\} and ε = \frac{1}{4}, where z ∈ C(S^{1}) is a canonical generator. One can prove that there is no integer L which is good for all (ϕ_{n}, ψ_{n}) as in Theorem 5.18, by using the variation of determinant.

5.20. If X is any finite CW complex such that K_{1}(C(X)) is a torsion group, then Theorem 5.18 holds for X — one needs to choose P_{A} accordingly, which is described below. Suppose that m_{1}, m_{2}, ⋯, m_{i} are the degrees of all the torsion elements in K_{0}(A) and K_{1}(A). Let m be the least common multiple of m_{1}, m_{2}, ⋯, m_{i}. Since K_{1}(A) is a torsion group, similar to the discussion in 5.13, an element α ∈ KK(A, B) is completely determined by the K-theory maps

α^{0} : K_{0}(A) ⟶ K_{0}(B),
α_{p}^{0} : K_{0}(A, ZZ/p) ⟶ K_{0}(B, ZZ/p),
α_{p}^{1} : K_{1}(A, ZZ/p) ⟶ K_{1}(B, ZZ/p),

where p are all the numbers with p|m. (In particular, the map α^{1} : K_{1}(A) ⟶ K_{1}(B) is determined by the above maps.) One can choose P to be a finite set of projections in M_{•}(A) ⋃ ⋃_{p|m} M_{•}(A ⊗ C(W_{p} ⊗ S^{1})) such that the set P\underline{K}(A) (defined in 5.16) generates a sub group containing the group

K_{0}(A) ⊕ ⨁_{k|m} K_{∗}(A, ZZ/k).

Similar to the proof of Theorem 5.18, we can prove the following theorem, since, to determine a KK-element α ∈ KK(A, B), one does not need the map from K_{1}(A). (G(P) and δ(P) can be chosen accordingly as in 5.17.)

Theorem 5.21. Suppose that X is a finite CW complex with K_{1}(X) a torsion group. Suppose that A = P M_{l}(C(X))P and P are as in 5.20. For any finite set F ⊂ A, positive number ε > 0, and positive integer M, there are a finite set G ⊂ A (G ⊃ G(P) large enough ), a positive number δ > 0 (δ < δ(P) small enough), and a positive integer L (large enough) such that the following statement is true. If ϕ, ψ ∈ Map(A, B) are G(P)-δ(P) multiplicative and

ϕ_{∗} = ψ_{∗} : P\underline{K}(A) ⟶ \underline{K}(B),

where B = QM_{•}(C(Y))Q with dim(Y) ≤ M, then there is a homomorphism ν ∈ Hom(A, M_{L}(B)) with finite dimensional image, and there is a unitary u ∈ M_{L+1}(B) such that

∥u(ϕ ⊕ ν)(a)u^{∗} − (ψ ⊕ ν)(a)∥ < ε

for all a ∈ F.

The following is a direct consequence of Theorem 5.18.

Corollary 5.22. Let A = C(X), where X is one of the spaces: [0, 1], S^{2}, T_{II,k} or T_{III,k}, and let P be as in 5.16. For any finite set F ⊂ C(X), any positive number ε > 0 and any positive integer M, there are a finite set G ⊂ C(X) (G ⊃ G(P) large enough), positive numbers δ > 0 (δ ≤ δ(P) small enough) and η > 0 (small enough) such that the following statement is true. Let B = M_{•}(C(Y)) with dim(Y) ≤ M, and p ∈ B, a projection. If ϕ, ψ ∈ Map(C(X), pBp) are G-δ multiplicative maps inducing the same maps ϕ_{∗} = ψ_{∗} : P\underline{K}(C(X)) ⟶ \underline{K}(B), and \{x_{1}, x_{2}, ⋯, x_{n}\} is an η-dense subset of X, and q_{1}, q_{2}, ⋯, q_{n} are mutually orthogonal projections in (1 − p)B(1 − p) with rank(q_{i}) ≥ rank(p), then there is a unitary

u ∈ (p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})B(p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})

such that

∥ϕ(f) ⊕ ∑_{i=1}^{n} f(x_{i})q_{i} − u (ψ(f) ⊕ ∑_{i=1}^{n} f(x_{i})q_{i}) u^{∗}∥ < ε, ∀f ∈ F.

In particular, if ψ is a homomorphism, then there is a homomorphism ϕ̃ ∈ Hom(C(X), (p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})B(p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})) (defined by ϕ̃(f) = u (ψ(f) ⊕ ∑_{i=1}^{n} f(x_{i})q_{i}) u^{∗}) such that

∥ϕ̃(f) − (ϕ(f) ⊕ ∑_{i=1}^{n} f(x_{i})q_{i})∥ < ε, ∀ f ∈ F.

Proof: Since X is not the space of a single point, we can assume that X, as a metric space, satisfies that diameter(X) = 1. Apply Theorem 5.18 to the finite set F ⊂ A, the positive number \frac{ε}{3} and the integer M to obtain G, δ, L as in Theorem 5.18. Choose a positive number η < \frac{1}{8ML^{2}} such that if dist(x, x′) < 8ML^{2} · η, then ∥f(x) − f(x′)∥ < \frac{ε}{3} for all f ∈ F. Let {x_{1}, x_{2}, ⋯, x_{n}} be an η-dense subset of X and let q_{1}, q_{2}, ⋯, q_{n} ∈ (1 − p)B(1 − p) be mutually orthogonal projections with rank(q_{i}) ≥ rank(p). Similar to the proof of Corollary 1.6.13, one can find a 8ML · η-dense subset {x_{k_{1}}, x_{k_{2}}, ⋯, x_{k_{l}}} ⊂ {x_{1}, x_{2}, ⋯, x_{n}} and mutually orthogonal projections Q_{1}, Q_{2}, ⋯, Q_{l} with rank(Q_{j}) ≥ ML · rank(p), such that

∥ ∑_{i=1}^{n} f(x_{i})q_{i} − ∑_{j=1}^{l} f(x_{k_{j}})Q_{j}∥ < \frac{ε}{3}, ∀f ∈ F.

Since rank(Q_{i}) ≥ ML · rank(p), it follows that [Q_{i}] ≥ L · [p]. Again, similar to the proofs of Corollaries 1.6.12 and 1.6.13, it can be proved that a homomorphism ν ∈ Hom(A, M_{L}(pBp)) with finite dimensional image (from 5.18) can be perturbed, at the expense of at most \frac{ε}{3} on the finite set F, to a homomorphism ν′ which is of the form

ν′(f) = ∑_{j=1}^{l} f(x_{k_{j}})q′_{j}

with [q′_{j}] ≤ [Q_{j}] (some of the projections q′_{j} could be zero). Hence the corollary follows (see the proof of Corollary 1.6.12).

⊓ ⊔

By the discussion in 1.2.19, we have the following corollary.

Corollary 5.23. Let A = M_{l}(C(X)), where X is one of the spaces: [0, 1], S^{2}, T_{II,k} or T_{III,k}, and let P be as in 5.16. For any finite set F ⊂ A, any positive number ε > 0 and any positive integer M, there are a finite set G ⊂ A (G ⊃ G(P) large enough), numbers δ > 0 (δ ≤ δ(P) small enough) and η > 0 (small enough) such that the following statement is true. Let B = M_{•}(C(Y)) with dim(Y) ≤ M, and p ∈ B a projection. If ϕ, ψ ∈ Map(A, pBp) are G-δ multiplicative maps inducing the same map ϕ_{∗} = ψ_{∗} : P\underline{K}(A) ⟶ \underline{K}(B), and \{x_{1}, x_{2}, ⋯, x_{n}\} is an η-dense subset of X, and

q_{1} = \underbrace{q′_{1} ⊕ q′_{1} ⊕ ⋯ ⊕ q′_{1}}_{l}, q_{2} = \underbrace{q′_{2} ⊕ q′_{2} ⊕ ⋯ ⊕ q′_{2}}_{l}, ⋯, q_{n} = \underbrace{q′_{n} ⊕ q′_{n} ⊕ ⋯ ⊕ q′_{n}}_{l}

are mutually orthogonal projections in (1 − p)B(1 − p) with rank(q_{i}) ≥ rank(p), then there is a unitary

u ∈ (p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})B(p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})

such that

∥ϕ(f) ⊕ ∑_{i=1}^{n} q′_{i} ⊗ f(x_{i}) − u (ψ(f) ⊕ ∑_{i=1}^{n} q′_{i} ⊗ f(x_{i})) u^{∗}∥ < ε, ∀f ∈ F.

In particular, if ψ is a homomorphism, then there is a homomorphism ϕ̃ ∈ Hom(C(X), (p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})B(p ⊕ q_{1} ⊕ q_{2} ⊕ ⋯ ⊕ q_{n})) such that

∥ϕ̃(f) − (ϕ(f) ⊕ ∑_{i=1}^{n} q′_{i} ⊗ f(x_{i}))∥ < ε, ∀ f ∈ F.

Proof: Thanks to Lemma 1.6.8, we can always assume that the two maps ϕ and ψ satisfy the condition that ϕ|_{M_{l}(C^{|} )} and ψ|_{M_{l}(C^{|} )} are homomorphisms. Using the condition ϕ_{∗} = ψ_{∗} : P\underline{K}(A) ⟶ \underline{K}(B), we can assume

ϕ|_{M_{l}(C^{|} )} = ψ|_{M_{l}(C^{|} )}

after conjugating with a unit. Now the corollary follows from the following claim. Claim: For any finite set F ⊂ A = M_{l}(C(X)), any ε > 0, there are a finite set G ⊂ A and a positive number δ > 0 such that if a map ϕ : A → B is G-δ multiplicative and ϕ|_{M_{l}(C^{|} )} is a homomorphism, then there are a map ϕ_{1} : C(X) → ϕ(e_{11})Bϕ(e_{11}) and an identification of ϕ(1)Bϕ(1) ≅ M_{l}(ϕ(e_{11})Bϕ(e_{11})) such that

∥ϕ(f) − (ϕ_{1} ⊗ 1_{l})(f)∥ < ε ∀f ∈ F.

Furthermore, if G_{1} ⊂ C(X) and δ_{1} > 0 are a pregiven finite set and a pregiven positive number, then one can modify the set G and the number δ so that the map ϕ_{1} above can be chosen to be G_{1}-δ_{1} multiplicative. Proof of Claim: Suppose 1 ∈ F. Let F_{1} = {a_{ij}|(a_{ij})_{l×l} ∈ F }(⊂ C(X)) be the set of all entries of the elements in F. Let G = {(b_{ij})_{l×l} = ∑ b_{ij}e_{ij} | b_{ij} ∈ F_{1} ∪ G_{1} ⊂ C(X)\}( ⊂ A) and δ = min (\frac{ε}{2l^{2}} , δ_{1}). Suppose that ϕ : M_{l}(C(X)) → B is G-δ multiplicative. Let

ϕ_{1} = ϕ|_{e_{11}M_{l}(C(X))e_{11}} : C(X) → ϕ(e_{11})Bϕ(e_{11}).

Obviously the G-δ multiplicativity of ϕ implies the G_{1}-δ_{1} multiplicativity of ϕ_{1}. Identify ϕ(1)Bϕ(1) ≅ (ϕ(e_{11})Bϕ(e_{11})) ⊗ M_{l} by sending ϕ(e_{ij}) to e_{ij} ∈ M_{l} ⊂ (ϕ(e_{11})Bϕ(e_{11})) ⊗ M_{l}. Under this identification, we have

(ϕ_{1} ⊗ 1_{l})(a) = ∑_{i,j} ϕ(e_{i1})ϕ_{1}(a_{ij})ϕ(e_{1j}),

where a = (a_{ij})_{l×l} ∈ M_{l}(C(X)). On the other hand, writing a = ∑ e_{1i}(a_{ij}e_{11})e_{1j} and using the G-δ multiplicativity of ϕ, we have

∥ϕ(a) − (ϕ_{1} ⊗ 1_{l})(a)∥ ≤ ∑_{i,j} ∥ϕ(e_{1i}(a_{ij}e_{11})e_{1j}) − ϕ(e_{i1})ϕ_{1}(a_{ij})ϕ(e_{1j})∥

= ∑_{i,j} ∥ϕ(e_{1i}(a_{ij}e_{11})e_{1j}) − ϕ(e_{i1})ϕ(a_{ij}e_{11})ϕ(e_{1j})∥

≤ ∑_{i,j} 2δ = 2l^{2}δ ≤ ε.

This proves the Claim. Applying the Claim, one can reduce the proof to the case A = C(X) which is Corollary 5.22.

⊓ ⊔

Definition 5.24. Let A be a unital C^{∗}-algebra, let

P ⊂ M_{•}(A) ∪ M_{•}(A ⊗ C(S^{1})) ∪ ⋃_{k=2}^{∞} M_{•}(A ⊗ C(W_{k} × S^{1}))

be a finite set of projections, and let G(P), δ(P) be as in 5.17. A G(P) − δ(P) multiplicative map ϕ : A → B is called quasi-P\underline{K}-homomorphism if there is a homomorphism ψ : A → B with ϕ(1_{A}) = ψ(1_{A}) such that

[ϕ]_{∗} = [ψ]_{∗} : P\underline{K}(A) → \underline{K}(B).

Using the above definition and Definition 4.38, we can restate the second part of Corollary 5.23 as below.

Lemma 5.25. Let A = M_{l}(C(X)), where X is one of the spaces: [0, 1], S^{2}, T_{II,k} or T_{III,k}, and let P be as in 5.16. For any finite set F ⊂ A, any positive number ε > 0 and any positive integer M, there are a finite set G ⊂ A (G ⊃ G(P) large enough), positive numbers δ > 0 (δ ≤ δ(P) small enough) and η > 0 (small enough) such that the following statement is true. Let B = M_{•}(C(Y)) with dim(Y) ≤ M, and let p ∈ B be a projection. If ϕ ∈ Map(A, pBp) is a G-δ multiplicative quasi-P\underline{K}-homomorphism, and λ ∈ Hom(A, (1 − p)B(1 − B)) has the property PE(rank(p), η), then there is a homomorphism ϕ̃ ∈ Hom(A, B) such that

∥ϕ̃(f) − (ϕ ⊕ λ)(f)∥ < ε, ∀f ∈ F.

Furthermore if Y is a connected simplicial complex different from the single point space, then ϕ̃ can be chosen to be injective.

Proof: The main body of the lemma is a restatement of Corollary 5.23. So we only need to prove the last sentence of the lemma. We need the following fact: Let X = [0, 1], S^{2}, T_{II,k} or T_{III,k}, and let Y be a connected finite simplicial complex different from {pt}. If λ_{1} : M_{l}(C(X)) → p_{1}M_{•}(C(Y))p_{1} is a homomorphism defined by the point evaluation at a point x_{1} ∈ X as

M_{l}(C(X)) ⟶^{e_{x_{1}}} M_{l}(C^{|} ) ⟶ p_{1}M_{•}(C(Y))p_{1},

then λ_{1} is homotopic to an injective homomorphism λ′_{1} : M_{l}(C(X)) → p_{1}M_{•}(C(Y))p_{1}. (Again, this fact can be proved by using the Peano Curve.) Let η′ be as the η desired in the main body of the lemma for \frac{ε}{2} (in place of ε). We can also assume that η′ satisfies the condition that if dist(x, x′) < η′, then ∥f(x) − f(x′)∥ < \frac{ε}{2} for all f ∈ F. Choose η = \frac{η′}{4} . Suppose that λ ∈ Hom(A, (1 − p)B(1 − p)) has the property PE(rank(p), η). Write λ = ⨁_{i=1}^{n} λ_{i}, where

λ_{i} : M_{l}(C(X)) ⟶^{e_{x_{i}}} M_{l}(C^{|} ) ⟶^{ϕ_{i}} p_{i}Bp_{i},

are point evaluations at an η-dense set {x_{1}, x_{2}, ⋯, x_{n}} and ϕ_{i} are unital homomorphisms. Let p_{1} be a projection with minimum rank among all the projections p_{1}, p_{2}, ⋯, p_{n}. Let ϕ ∈ Map(A, pBp) be a G-δ multiplicative quasi-P\underline{K}- homomorphism. Then ϕ ⊕ λ_{1} ∈ Map(A, (p ⊕ p_{1})B(p ⊕ p_{1})) is also a G-δ multiplicative quasi-P\underline{K}-homomorphism. Furthermore, from the above fact, it defines the same map on the level of P\underline{K}(A) as an injective homomorphism ψ ∈ Hom(A, (p ⊕ p_{1})B(p ⊕ p_{1})). On the other hand, λ′ = ⨁_{i=2}^{n} λ_{i} has the properties PE(rank(p), 2η) and PE(rank(p_{1}), 2η), since λ = ⨁_{i=1}^{n} λ_{i} has PE(rank(p), η), and rank(p_{1}) ≤ rank(p_{i}), i = 2, ⋯, n. Similar to the proof of Corollary 5.22, λ′ can be perturbed to a homomorphism λ′′ which has the property PE(rank(p) + rank(p_{1}), 4η) at the expense of at most \frac{ε}{2} on the finite set F. Note that 4η = η′, and ψ is injective. Hence the homomorphism Adu ∘ (ψ ⊕ λ′′) (for a certain unitary u), as desired in the main body of the lemma, is also injective.

⊓ ⊔

Lemma 5.26. Let X and Y be connected finite simplicial complexes. Suppose that ϕ_{1} : P M_{k}(C(X))P → Q_{1}M_{l}(C(Y))Q_{1} and ϕ_{2} : P M_{k}(C(X))P → Q_{2}M_{l}(C(Y))Q_{2} are unital homomorphisms, where P, Q_{1}, and Q_{2} are projections with

rank(Q_{2}) − rank(Q_{1}) ≥ 2 dim(Y) · rank(P).

Then there exists a homomorphism ψ : P M_{k}(C(X))P → M_{•}(C(Y)) such that

[ψ] = [ϕ_{2}] − [ϕ_{1}] ∈ KK(C(X), C(Y)).

Proof: First, we suppose that A = C(X). As in Lemma 3.14 of [EG 2] (see Remark 1.6.21 above), we can assume that ϕ_{1}(C_{0}(X)) ⊂ M_{l}(C_{0}(Y)) and ϕ_{2}(C_{0}(X)) ⊂ M_{l}(C_{0}(Y)), where C_{0}(X) and C_{0}(Y) are sets of functions vanishing on fixed base points of X and Y, respectively. Hence ϕ_{i} defines an element kk(ϕ_{i}) ∈ kk(Y, X) (see [DN]). Furthermore, [ϕ_{i}] ∈ KK(C(X), C(Y)) is completely determined by kk(ϕ_{i}) and ϕ_{i∗}([1_{A}]) ∈ K_{0}(B). Let α = kk(ϕ_{2}) − kk(ϕ_{1}) ∈ kk(Y, X) (note that kk(Y, X) is an abelian group, see [DN]). Since rank(Q_{2})−rank(Q_{1}) ≥ 2 dim(Y), by [Hu], there is a projection Q_{3} ∈ M_{•}(C(Y)) such that [Q_{3}] = [Q_{2}] − [Q_{1}] ∈ K_{0}(C(Y)). By Theorem 4.11 of [DN] or Lemma 3.16 of [EG2], there is a unital homomorphism ψ : C(X) → Q_{3}M_{•}(C(Y))Q_{3} to realize α ∈ kk(Y, X). Obviously ψ is as desired. For the general case, using the Dilation Lemma (Lemma 1.3.1), one can prove that [ϕ_{i}] ∈ KK(C(X), C(Y)) can be realized by homomorphism ϕ′_{i} : C(X) → M_{•}(C(Y)). This reduces the proof to the above case.

⊓ ⊔

Remark 5.27. In the above lemma, if Q_{1} < Q_{2}, then one can choose ψ to satisfy ψ(1_{A}) = Q_{2} − Q_{1}.

Lemma 5.28. Let X be a finite simplicial complex, and A = P M_{l}(C(X))P.

For any finite set

P ⊂ M_{•}(A) ∪ M_{•}(A ⊗ C(S^{1})) ∪ ⋃_{k=2}^{∞} M_{•}(A ⊗ C(W_{k} × S^{1})),

there are a finite set G ⊂ A and a number δ > 0, such that the following is true. If Y is a simplicial complex, Q > Q_{1} are two projections in M_{•}(C(Y)) with rank(Q) − rank(Q_{1}) ≥ 2 dim(Y)rank(P), and two unital homomorphisms ϕ ∈ Hom(A, QM_{•}(C(Y))Q)_{1}, ϕ_{1} ∈ Hom(A, Q_{1}M_{•}(C(Y))Q_{1})_{1} and a unital map ϕ_{2} ∈ Map(A, (Q − Q_{1})M_{•}(C(Y))(Q − Q_{1}))_{1} satisfy that

(∗) ∥ϕ(f) − ϕ_{1}(f) ⊕ ϕ_{2}(f)∥ < δ, ∀g ∈ G,

then there is a homomorphism ψ : A → (Q − Q_{1})M_{•}(C(Y))(Q − Q_{1}) such that

[ψ]_{∗} = [ϕ_{2}]_{∗} : P\underline{K}(A) → \underline{K}(C(Y)).

In other words, ϕ_{2} is a quasi-P\underline{K}-homomorphism. (Notice that, from Lemma 4.40, if G is large enough and δ is small enough, then (*) above implies that ϕ_{2} ∈ Map(A, (Q − Q_{1})M_{•}(C(Y))(Q − Q_{1}))_{1} is G(P) − δ(P) multiplicative, and hence [ϕ_{2}]_{∗} : P\underline{K}(A) → \underline{K}(C(Y)) makes sense.)

Proof: If G is large enough and δ is small enough, then (*) implies

[ϕ_{2}]_{∗} = [ϕ]_{∗} − [ϕ_{1}]_{∗} : P\underline{K}(A) → \underline{K}(C(Y)).

Then the lemma follows from Lemma 5.26 and Remark 5.27.

⊓ ⊔

Remark 5.29. In Corollary 4.39, we can choose ψ_{0} (or ψ′_{0}) such that ψ_{0}^{i,j} (or ψ_{0}^{′i,j}) is a quasi-P\underline{K}-homomorphism for any pre-given set of projections

P ⊂ M_{•}(A) ∪ M_{•}(A ⊗ C(S^{1})) ∪ ⋃_{k=2}^{∞} M_{•}(A ⊗ C(W_{k} × S^{1})).

To do so, by Lemma 5.28, one only needs to choose the projection Q_{0}^{i,j} to have rank at least 2 dim(X_{m,j}) · rank(1_{A_{n}^{i}}). But from the construction in 4.34, we have freedom to do so.

Lemma 5.30. Fix a positive integer M. Suppose that B = ⨁_{i=1}^{s} M_{l_{i}}(C(Y_{i})), where Y_{i} are the spaces: {pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k}, S^{2}. For any finite set G ⊂ B and positive number ε > 0, there exist a finite set G_{1} ⊂ B, numbers δ_{1} > 0 and η > 0 such that the following is true. If a map α = α_{0} ⊕ α_{1} : B → A = ⨁_{j=1}^{t} M_{k_{j}}(C(X_{j})), with dim(X_{j}) ≤ M, satisfies the following conditions:

(1) α_{0} is G_{1}-δ_{1} multiplicative, {α_{0}(1_{B^{i}})}_{i=1}^{s} are mutually orthogonal projections, and α_{1} is a homomorphism with finite dimensional image (i.e., defined by point evaluations);

(2) For any block B^{i} with Y_{i} = T_{II,k}, T_{III,k} or S^{2} and any block A^{j}, the partial map α_{0}^{i,j} is quasi-P\underline{K}-homomorphism, where P is the set of projections associated to B^{i} as in 5.16, and the homomorphism α_{1}^{i,j} has the property PE(rankα_{0}^{i,j}(1_{B^{i}}), η);

then there is a unital homomorphism α′ : B → α(1_{B})Aα(1_{B}) such that

∥α′(g) − α(g)∥ < ε, ∀g ∈ G.

Proof: We only need to perturb all the individual maps α^{i,j} to homomorphisms α^{′i,j} within α^{i,j}(1_{B^{i}})A^{j}α^{i,j}(1_{B^{i}}). For a block of B^{i} with spectrum {pt}, [0, 1] or S^{1}, such perturbation exists by Lemma 1.6.1. For a block of B^{i} with spectrum T_{II,k}, T_{III,k} or S^{2}, such perturbation exists by Lemma 5.25.

⊓ ⊔

Lemma 5.31. Let M be a fixed positive integer. Let B = M_{l}(C(Y)), Y = T_{II,k}, T_{III,k} or S^{2}. Let the set of projections P ⊂ M_{•}(B)∪M_{•}(B⊗C(W_{k}×S^{1})) be as in 5.16. Let A = RM_{l_{1}}(C(X))R with dim(X) ≤ M, where R ∈ M_{l_{1}}(C(X)) is a projection. Let α : B → A be an injective homomorphism. Let a finite set of projections P′ be given by P′ := (α ⊗ id)(P) ⊂ M_{•}(A) ∪ M_{•}(A ⊗ C(W_{k} × S^{1})). Let η > 0. Choose η_{1} > 0 such that if a finite set {x_{1}, x_{2}, ⋯, x_{n}} ⊂ X is η_{1}-dense in X, then ⋃_{i=1}^{n} SPα_{x_{i}} is η-dense in Y. (Such η_{1} exists because of injectivity of α.) For any finite subset G_{1} ⊂ B and any number δ_{1} > 0, there are a finite subset G_{2} ⊂ A and a number δ_{2} > 0 such that the following are true. Let C = M_{•}(Z) with dim(Z) ≤ M.

(1) If ψ_{0} : A → Q_{0}CQ_{0} is a G_{2}-δ_{2} multiplicative quasi-P′\underline{K}-homomorphism and ψ_{0}(α(1_{B})) is a projection, then ψ_{0} ∘ α is a G_{1}-δ_{1} multiplicative quasi-P\underline{K}- homomorphism.

(2) If ψ_{1} : A → Q_{1}CQ_{1} has the property PE(J · L, η_{1}), where J = rank(R), then ψ_{1} ∘ α : B → ψ_{1}(α(1_{B}))Cψ_{1}(α(1_{B})) has property PE(L, η). In particular, if ψ_{1} has the property PE(J ·rank(Q_{0}), η_{1}), (this is the condition (3) of Corollary 4.39), then ψ_{1} ∘ α has the property PE(rank((ψ_{0} ∘ α)(1_{B})), η). (Note that rank((ψ_{0} ∘α)(1_{B})) ≤ rank(Q_{0}).) Consequently, if we further assume that G_{1}, δ_{1} and η are as those chosen in 5.30 for a finite set G ⊂ B and ε > 0, and Q_{1} is orthogonal to Q_{0}, then there is a homomorphism ψ : B → (Q_{0} ⊕ Q_{1})C(Q_{0} ⊕ Q_{1}) such that

∥ψ(g) − (ψ_{0} ⊕ ψ_{1})(α(g))∥ < ε, ∀g ∈ G.

Proof: (1) holds if we choose G_{2} ⊃ α(G) and δ_{2} < δ_{1}.

(2) follows from the following fact: if a homomorphism ϕ : RM_{l_{1}}(C(X))R → M_{t}(C(Z)) contains a part of point evaluation at a point x ∈ X of size at least L (see Definition 4.38), then for any y ∈ SPα_{x} ⊂ Y, ϕ ∘ α contains a part of point evaluation at point y of size at least \frac{L}{rank(R)}.

⊓ ⊔

The following two theorems are important for the proof of our main theorem.

Theorem 5.32a.  Let M be a positive integer.  Let

lim_{n→∞}(A_{n} = ⨁_{i=1}^{k_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) be a simple inductive limit with injective connecting homomorphisms ϕ_{n,m} and with dim(X_{n,i}) ≤ M, for any n, i. Let B = ⨁_{i=1}^{s} M_{l_{i}}(C(Y_{i})), where Y_{i} are the spaces: {pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k}, and S^{2}. Suppose that a homomorphism α : B → A_{n} satisfies the following dichotomy condition: For any block B^{i} of B and any block A_{n}^{j} of A_{n}, either the partial map α^{i,j} : B^{i} → A_{n}^{j} is injective or it has a finite dimensional image. Denote α(1_{B}) := R(= ⨁ R^{i}) ∈ A_{n}(= ⨁ A_{n}^{i}). For any finite sets G ⊂ B and F ⊂ RA_{n}R, any positive number ε > 0, and any positive integer L, there are A_{m} and mutually orthogonal projections Q_{0}, Q_{1}, Q_{2} ∈ A_{m}, with ϕ_{n,m}(R) = Q_{0} + Q_{1} + Q_{2}, a unital map θ_{0} ∈ Map(RA_{n}R, Q_{0}A_{m}Q_{0})_{1}, two unital homomorphisms θ_{1} ∈ Hom(RA_{n}R, Q_{1}A_{m}Q_{1})_{1} and ξ ∈ Hom(RA_{n}R, Q_{2}A_{m}Q_{2})_{1} such that

(1) ∥ϕ_{n,m}(f) − (θ_{0}(f) + θ_{1}(f) + ξ(f))∥ < ε, ∀f ∈ F;

(2) there is a homomorphism α_{1} : B → (Q_{0} ⊕ Q_{1})A_{m}(Q_{0} ⊕ Q_{1}) such that

∥α_{1}(g) − (θ_{0} + θ_{1}) ∘ α(g)∥ < ε, ∀g ∈ G;

(3) θ_{0} is F − ε multiplicative and θ_{1} satisfies that for any nonzero projection (including any rank 1 projection) e ∈ R^{i}A_{n}^{i}R^{i}

θ_{1}^{i,j}([e]) ≥ L · [θ_{0}^{i,j}(R^{i})],

(the condition (3) will be used when we apply Theorem 1.6.9 in the proof of the Main Theorem);

(4) ξ factors through a C^{∗}-algebra C—a direct sum of matrix algebras over C[0, 1] or C^{|} — as

ξ : RA_{n}R ⟶^{ξ_{1}} C ⟶^{ξ_{2}} Q_{2}A_{m}Q_{2},

and the partial maps of ξ_{2} satisfy the dichotomy condition;

(5) the partial maps of α_{1} satisfies the dichotomy condition.

Proof: Let E^{i,j} = α^{i,j}(1_{B^{i}}) ∈ A_{n}^{j}. Let

I = {(i, j) | α^{i,j} : B^{i} → A_{n}^{j} has finite dimensional image}.

Let the subalgebra D ⊂ A_{n} = ⨁ A_{n}^{i} be defined by

D = ⨁_{j} ( ⨁_{(i,j)∈I} α^{i,j}(B^{i}) ⊕ ⨁_{(i,j)∉I} α^{i,j}(C^{|} · 1_{B^{i}})) ⊂ ⨁_{j} A_{n}^{j}.

Notice that D is a finite dimensional subalgebra of A_{n} containing the mutually orthogonal projections {E^{i,j} = α^{i,j}(1_{B_{i}})}_{i,j}. Apply part 2 of Corollary 4.39 for sufficiently large set F′ ⊂ RA_{n}R, sufficiently small number ε′ > 0 and η′ > 0, and positive integer J = L · max_{i} rank(R^{i}), to obtain A_{m} and the decomposition θ_{0}⊕θ_{1}⊕ξ of ϕ_{n,m}|_{RA_{n}R} as ψ′_{0}⊕ψ′_{1}⊕ψ′_{2} in 4.39. By Lemma 1.6.8, we can assume that the restriction θ_{0}|_{D} is a homomorphism. The condition (1) follows if we choose F′ ⊃ F, and ε′ < ε. The F − ε multiplicativity of θ_{0} in (3) follows from Lemma 4.40, if F′ is large enough and ε′ is small enough, and the desired property of θ_{1} in (3) follows from the choice of J and Lemma 5.31. To construct α_{1} as desired in the condition (2), we need to construct

α_{1}^{i,j,k} : B^{i} → θ^{j,k}(E^{i,j})A_{m}^{k}θ^{j,k}(E^{i,j}),

where θ = θ_{0} ⊕ θ_{1}, to satisfy

∥α_{1}^{i,j,k}(g) − θ^{j,k} ∘ α^{i,j}(g)∥ < ε, ∀g ∈ G.

The construction are divided into three cases.

1. If (i, j) ∈ I, then θ^{j,k} ∘ α^{i,j} is already a homomorphism and can be chosen to be α_{1}^{i,j,k}.

2. If (i, j) ∉ I, and Y_{i} = [0, 1] or S^{1}, then the existence of α_{1}^{i,j,k} follows from Lemma 1.6.1 and Lemma 4.40, if F′ is large enough and ε′ is small enough. (See Lemma 5.30 also.) In fact, in this case, the map θ_{0}^{j,k} ∘ α^{i,j} itself can be perturbed to a homomorphism. On the other hand, the homomorphism θ_{1}^{j,k} ∘ α^{i,j} is defined by the point evaluations on an η-dense set for a certain small number η. Evidently, such a homomorphism θ_{1}^{j,k} ∘ α^{i,j} from M_{l_{i}}(C(S^{1})) or M_{l_{i}}(C([0, 1])) (to A_{m}^{k}) can be perturbed to an injective homomorphism, provided that η is sufficiently small and that the path connected simplicial complex X_{m,k} is not the space of a single point. Therefore, in this case, the homomorphism α_{1}^{i,j,k} can be chosen to be injective.

3. If (i, j) ∉ I, and Y_{i} = T_{II,k}, T_{III,k} or S^{2}, then α^{i,j} is injective, and the existence of α_{1}^{i,j,k} follows from Lemma 5.30 and the choice of J, if F′ is large enough and ε′ is small enough, and if we choose η′ to be the number η_{1} in Lemma 5.31 corresponding to the η in Lemma 5.30. The homomorphism α_{1}^{i,j,k} can also be chosen to be injective, if X_{m,k} is not the space of a single point, according to the last part of Lemma 5.25. Finally, define the partial map α_{1}^{i,k} of α_{1} to be ⨁_{j} α_{1}^{i,j,k} to complete the construction. Obviously, it follows, from the discussion of the injectivity in case 2 and case 3, that α_{1} satisfies the dichotomy condition.

⊓ ⊔

Theorem 5.32b.  Let M be a positive integer.  Let

lim_{n→∞}(A_{n} = ⨁_{i=1}^{k_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) be a simple inductive limit with injective connecting homomorphisms ϕ_{n,m} and with dim(X_{n,i}) ≤ M, for any n, i. Let B = ⨁_{i=1}^{s} M_{l_{i}}(C(Y_{i})), where Y_{i} are the spaces:{pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k}, and S^{2}. Suppose that a homomorphism α : B → A_{n} satisfies the following dichotomy condition: For any block B^{i} of B and any block A_{n}^{j} of A_{n}, either the partial map α^{i,j} : B^{i} → A_{n}^{j} is injective or it has a finite dimensional image. For any finite sets G ⊂ B and F ⊂ A_{n}, and any number ε > 0, there are A_{m} and mutually orthogonal projections P, Q ∈ A_{m}, with ϕ_{n,m}(1_{A_{n}}) = P + Q, a unital map θ ∈ Map(A_{n}, P A_{m}P)_{1}, and a unital homomorphism ξ ∈ Hom(A_{n}, QA_{m}Q)_{1} such that

(1) ∥ϕ_{n,m}(f) − (θ(f) ⊕ ξ(f))∥ < ε, ∀f ∈ F;

(2) there is a homomorphism α_{1} : B → P A_{m}P such that

∥α_{1}(g) − (θ ∘ α)(g)∥ < ε, ∀g ∈ G;

(3) θ(F) is weakly approximately constant to within ε;

(4) ξ factors through a C^{∗}-algebra C—a direct sum of matrix algebras over C[0, 1] or C^{|} — as

ξ : A_{n} ⟶^{ξ_{1}} C ⟶^{ξ_{2}} QA_{m}Q,

and the partial maps of ξ_{2} satisfy the dichotomy condition;

(5) the partial maps of α_{1} satisfy the dichotomy condition. The proof is similar to the proof of Theorem 5.32a, we omit it.

6 The proof of the main theorem

In this section, we will combine §4, §5 and §1.6 to prove our Main Theorem — the Reduction Theorem. The following is Proposition 3.1 of [D2].

Proposition 6.1. ([D2, 3.1]) Consider the diagram

A_{1}  ⟶^{ϕ_{1,2}} A_{2}  ⟶^{ϕ_{2,3}} ⋯ ⟶ A_{n} ⟶^{ϕ_{n,n+1}} A_{n+1} ⟶ ⋯
↑_{α_{1}}  ^{β_{1}} ↘  ↑_{α_{2}}  ^{β_{2}} ↘ ⋯  ↑_{αn}  ^{βn} ↘  ↑_{α_{n+1}} ↘
B_{1}  ⟶^{ψ_{1,2}} B_{2}  ⟶^{ψ_{2,3}} ⋯ ⟶ B_{n} ⟶^{ψ_{n,n+1}} B_{n+1} ⟶ ⋯,

where A_{n}, B_{n} are C^{∗}-algebras, ϕ_{n,n+1}, ψ_{n,n+1} are homomorphisms and α_{n}, β_{n} are linear ∗-contractions.

Suppose that F_{n} ⊂ A_{n}, E_{n} ⊂ B_{n} are finite sets satisfying the following conditions.

ϕ_{n,n+1}(F_{n}) ∪ α_{n+1}(E_{n+1}) ⊂ F_{n+1}, ψ_{n,n+1}(E_{n}) ∪ β_{n}(F_{n}) ⊂ E_{n+1},

and \overline{⋃_{n=1}^{∞}(ϕ_{n,∞}(F_{n}))} and \overline{⋃_{n=1}^{∞}(ψ_{n,∞}(E_{n}))} are the unit balls of A = lim(A_{n}, ϕ_{n,m}) and B = lim(B_{n}, ψ_{n,m}), respectively. Suppose that there is a sequence ε_{1}, ε_{2}, ⋯ of positive numbers with ∑ ε_{n} < +∞ such that α_{n} and β_{n} are F_{n}-ε_{n} multiplicative and E_{n}-ε_{n} multiplicative, respectively, and

∥ϕ_{n,n+1}(f) − α_{n+1} ∘ β_{n}(f)∥ < ε_{n}, and ∥ψ_{n,n+1}(g) − β_{n} ∘ α_{n}(g)∥ < ε_{n}

for all f ∈ F_{n} and g ∈ E_{n}. Then A is isomorphic to B.

Lemma 6.2. Let lim(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) be a simple inductive limit C^{∗}-algebra with ϕ_{n,m} injective, where X_{n,i} are path connected finite simplicial complexes with uniformly bounded dimensions. Let C^{∗}-algebra C be a direct sum of matrix algebras over the spaces: {pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k} and S^{2}, and ϕ : C → A_{n} be an injective homomorphism. Then for any finite set F ⊂ C and ε > 0, there is a positive integer N > n such that for any m > N, there is a homomorphism ψ : C → A_{m} satisfying the following conditions.

(1) ψ(1_{C^{i}}) = (ϕ_{n,m} ∘ ϕ)(1_{C^{i}}), for any block C^{i} of C.

(2) ∥ψ(f) − (ϕ_{n,m} ∘ ϕ)(f)∥ < ε, ∀f ∈ F.

(3) ψ satisfies the following dichotomy condition: For any block C^{i} of C and A_{m}^{j} of A_{m}, either ψ^{i,j} is injective or ψ^{i,j} has finite dimensional image. (For the proof of the main theorem of this article—Theorem 6.3 below, we only need this lemma for the case that C is a direct sum of matrix algebras over spaces {pt} and [0, 1]. The full generality of the lemma will be used in the proof of Corollary 6.11 below.)

Proof: We only need to prove for the case that C has only one block C = M_{k}(C(X)). And, by the discussion in 1.2.19, this case can further be reduced to the case C = C(X). For the finite set F ⊂ C, there is an η > 0 such that if dist(t, t′) < 4η, then

∥f(t) − f(t′)∥ < ε, ∀f ∈ F.

Let (X, σ) be a simplicial decomposition of X such that for any simplex Δ ⊂ (X, σ), diameter(Δ) < η. We call a simplex Δ a top simplex if Δ is not a proper face of any simplex. Obviously, Δ is a top simplex if and only if the interior Δ^{∘} is an open subset of X.

From [DNNP, Proposition 2.1], using the injectivity of ϕ and ϕ_{n,m}, it follows that there is an integer N > n such that for any open set Δ^{∘} — the interior of a top simplex Δ ⊂ (X, σ), one has

SP (ϕ_{n,m} ∘ ϕ)_{y} ∩ Δ^{∘} ≠ ∅.

We can define the homomorphism ψ : C → A_{m} for each block A_{m}^{j} of A_{m} separately. That is, we need to define ψ^{j} : C → A_{m}^{j}, then let ψ := ⊕ψ^{j}. If SP(A_{m}^{j}) = X_{m,j} = {pt}, then the partial map (ϕ_{n,m} ∘ ϕ)^{j} has finite dimensional image, and we can define it to be ψ^{j}. Hence we assume that the connected finite simplicial complex X_{m,j} is not the space of single point {pt}. Let α = (ϕ_{n,m} ∘ ϕ)^{j} : C → A_{m}^{j}.

Let Y be the union of all such top simplices Δ that Δ^{∘} ∩ SPα is uncountable.

Let Z be the union of all simplices Δ which are not top simplices. Both Y and Z are closed subset of X. Let Δ_{1}, Δ_{2}, ⋯, Δ_{l} be the list of all top simplices such that Δ_{i} ⊄ Y, i = 1, 2, ⋯, l. Then

X = Y ∪ Δ_{1} ∪ Δ_{2} ⋯ ∪ Δ_{l}.

(This fact will be used later.) (Here we use the fact that X is equal to the union of all top simplices, since each simplex is a face of a top simplex.) For each Δ_{i}, Δ_{i}^{∘} ∩ SPα is a countable nonempty set. There is a point x_{i} and

an open disk U_{i} = B_{ε_{i}}(x_{i}) ∋ x_{i} such that

SPα ∩ U_{i} = {x_{i}}.

We can assume that U_{i} ⊂ Δ_{i}^{∘}. Obviously, ∂Δ_{i} is a deformation retract of

Δ_{i}\U_{i}. Set (X\ (∪_{i=1}^{l}U_{i})) ∩ SPα = T. Then SPα = T ∪ {x_{1}, x_{2}, ⋯, x_{l}}. Define a function g : T → Y ∪ Z(⊂ X) as below. Let g′ : Z → Y ∪ Z be the identity map, that is,

g′(z) = z, ∀z ∈ Z.

We will extend the map g′ to a map (let us still denote it by g′).

g′ : X\ (∪_{i=1}^{l}U_{i}) ⟶ Y ∪ Z.

For each top simplex Δ ⊂ Y, extend g′|_{∂Δ} to a map g′ : Δ → Δ satisfying

g′(T ∩ Δ) = Δ.

(Such extension exists since T ∩ Δ^{∘} is uncountable, see Lemma 2.6 of [EGL].)

For any simplex Δ_{i}, i = 1, 2, ⋯, l, one can extend g′|_{∂Δ} to a map g′ : Δ_{i}\U_{i} → ∂Δ_{i}, since ∂Δ_{i} is a deformation retract of Δ_{i}\U_{i}.

Thus we obtain the extension g′ : X\ (∪_{i=1}^{l}U_{i}) ⟶ Y ∪ Z. Let g = g′|_{T}. Then

g(T) ⊃ Y, and dist(g(x), x) < η, ∀x ∈ T.

Since SPα = T ∪ {x_{1}, x_{2}, ⋯, x_{l}}, there are homomorphisms α_{0} : C(T) → A_{m}^{j} and α_{i} : C^{|} = C({x_{i}}) → A_{m}^{j}, j = 1, 2, ⋯, l, with mutually orthogonal images, such that

α(f) = α_{0}(f|_{T}) + ∑_{i=1}^{l}(f|_{{x_{i}}}), ∀f ∈ C(X).

Define β_{0} : C(Y ∪ Z) → A_{m}^{j} by

β_{0}(f) = α_{0}(f ∘ g), ∀f ∈ C(Y ∪ Z),

where g : T → Y ∪ Z is defined as above. For each Δ_{i}, there is a surjective map g_{i} : X_{m,j} → Δ_{i}, since X_{m,j} ≠ {pt}. Define β_{i} : C(Δ_{i}) → A_{m}^{j} by

β_{i}(f)(x) = f(g_{i}(x)) · α_{i}(1C_{|} ), ∀f ∈ C(Δ_{i}), x ∈ X_{m,j}.

Then, obviously, we have (1′) β_{0}(1_{C(Y ∪Z)}) = α_{0}(1_{C(T)}), and β_{i}(1_{C(Δ_{i})}) = α_{i}(1C_{|} ), for i = 1, 2, ⋯, l. From the way η is chosen and the properties that dist(g(x), x) < η for any x ∈ T, and that diameter(Δ_{i}) < η for any i = 1, 2, ⋯, l, we have (2′) ∥β_{0}(f|_{Y ∪Z}) − α_{0}(f|_{T})∥ < ε, and ∥β_{i}(f|_{Δ_{i}}) − α_{i}(f|_{{x_{i}}})∥ < ε for i = 1, 2, ⋯, l, and f ∈ F. Finally, let the partial homomorphism ψ^{j} : C(X) → A_{m}^{j} be defined by

ψ^{j}(f) = β_{0}(f|_{Y ∪Z}) + ∑_{i=1}^{l} β_{i}(f|_{Δ_{i}}).

Since T ⊂ SPα_{0} and the map g : T → Y ∪ Z satisfies g(T) ⊃ Y, we have SP(β_{0}) ⊃ Y. Hence SPψ^{j} = SPβ_{0} ∪ ∪_{i=1}^{l}SPβ_{i} ⊃ Y ∪ ∪_{i=1}^{l}Δ_{i} = X. That is, ψ^{j} is injective. The property (1) follows from (1′) and (2) follows from (2′).

⊓ ⊔

We will use 5.32a, 5.32b, 1.6.9, 1.6.29, 1.6.30 to prove the following main theorem of this article.

Theorem 6.3. Suppose that lim(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) is a simple inductive limit C^{∗}-algebra with dim(X_{n,i}) ≤ M for a fixed positive integer M. Then there is another inductive system (B_{n} = ⨁_{i=1}^{t_{n}} M_{{n,i}}(C(Y_{n,i})), ϕ_{n,m}) with the same limit algebra as the above system, where all Y_{n,i} are spaces of forms {pt}, [0, 1], S^{1}, S^{2}, T_{II,k}, or T_{III,k}.

Proof: Without loss of generality, assume that the spaces X_{n,i} are connected finite simplicial complexes and the connecting maps ϕ_{n,m} are injective (see Theorem 4.23).

Let ε_{1} > ε_{2} > ε_{3} > ⋯ > 0 be a sequence of positive numbers satisfying ∑ ε_{n} < +∞. We need to construct the intertwining diagram

⋂^{F_{1}}  ⋂^{F_{2}}  ⋂^{F_{n}}  ⋂^{F_{n+1}}
A_{s(1)} ⟶^{ϕ_{s(1),s(2)}} A_{s(2)} ⟶^{ϕ_{s(2),s(3)}} ⋯ ⟶ A_{s(n)} ⟶^{ϕ_{s(n),s(n+1)}} A_{s(n+1)} ⟶ ⋯
↑_{α_{1}}  ^{β_{1}} ↘  ↑_{α_{2}}  ^{β_{2}} ↘  ↑_{αn}  ^{βn} ↘  ↑_{α_{n+1}} ↘
B_{1}  ⟶^{ψ_{1,2}}  B_{2}  ⟶^{ψ_{2,3}} ⋯ ⟶ B_{n}  ⟶^{ψ_{n,n+1}}  B_{n+1} ⟶ ⋯
⋃_{E_{1}}  ⋃_{E_{2}}  ⋃_{E_{n}}  ⋃_{E_{n+1}}

satisfying the following conditions. (0.1) (A_{s(n)}, ϕ_{s(n),s(m)}) is a sub-inductive system of (A_{n}, ϕ_{n,m}). (B_{n}, ψ_{n,m}) is an inductive system of matrix algebras over the spaces: {pt}, [0, 1], S^{1}, {T_{II,k}}_{i=2}^{∞}, {T_{III,k}}_{i=2}^{∞}, S^{2}. (0.2) Choose {a_{ij}}_{j=1}^{∞} ⊂ A_{s(i)} and {b_{ij}}_{j=1}^{∞} ⊂ B_{i} to be countable dense subsets of the unit balls of A_{s(i)} and B_{i}, respectively. F_{n} are subsets of the unit balls of A_{s(n)}, and E_{n} are subsets of the unit balls of B_{n} satisfying

ϕ_{s(n),s(n+1)}(F_{n}) ∪ α_{n+1}(E_{n+1}) ∪ ⋃_{i=1}^{n+1} ϕ_{s(i),s(n+1)}({a_{i1}, a_{i2}, ⋯, a_{i n+1}}) ⊂ F_{n+1}

and

ψ_{n,n+1}(E_{n}) ∪ β_{n}(F_{n}) ∪ ⋃_{i=1}^{n+1} ψ_{i,n+1}({b_{i1}, b_{i2}, ⋯, b_{i n+1}}) ⊂ E_{n+1}.

(Here we use the convention that ϕ_{n,n} = id : A_{n} → A_{n}.) (0.3) β_{n} are F_{n} − 2ε_{n} multiplicative and α_{n} are homomorphisms. (0.4) ∥ψ_{n,n+1}(g) − β_{n} ∘ α_{n}(g)∥ < 2ε_{n} for all g ∈ E_{n}, and ∥ϕ_{s(n),s(n+1)}(f) − α_{n+1} ∘ β_{n}(f)∥ < 12ε_{n} for all f ∈ F_{n}.

(0.5) For any block B_{n}^{i} of B_{n} and any block A_{s(n)}^{j} of A_{s(n)}, the map α_{n}^{i,j} satisfies the following dichotomy condition: either α_{n}^{i,j} is injective or α_{n}^{i,j} has a finite dimensional image. The diagram will be constructed inductively. First, let B_{1} = {0}, A_{s(1)} = A_{1}, α_{1} = 0. Let b_{1j} = 0 ∈ B_{1} for j = 1, 2, ⋯, and let {a_{1j}}_{j=1}^{∞} be a countable dense subset of the unit ball of A_{s(1)}. And let E_{1} = {b_{11}} = B_{1} and F_{1} = {a_{11}} ⊂ A_{s(1)}.

As an inductive assumption, assume that we already have the diagram

⋂^{F_{1}}  ⋂^{F_{2}}  ⋂^{F_{n}}
A_{s(1)} ⟶^{ϕ_{s(1),s(2)}} A_{s(2)} ⟶^{ϕ_{s(2),s(3)}} ⋯ ⟶ A_{s(n)}
↑_{α_{1}}  ^{β_{1}} ↘  ↑_{α_{2}}  ^{β_{2}} ↘  ⋯ ^{β_{n−1}} ↘  ↑_{αn}
B_{1}  ⟶^{ψ_{1,2}}  B_{2}  ⟶^{ψ_{2,3}} ⋯ ⟶ B_{n}
⋃_{E_{1}}  ⋃_{E_{2}}  ⋃_{E_{n}}

and, for each i = 1, 2 ⋯, n, we have countable dense subsets {a_{ij}}_{j=1}^{∞} ⊂ unit ball of A_{s(i)} and {b_{ij}}_{j=1}^{∞} ⊂ unit ball of B_{i} to satisfy the conditions (0.1)- (0.5) above. We have to construct the next piece of the diagram,

F_{n} ⊂ A_{s(n)} ⟶^{ϕ_{s(n),s(n+1)}} A_{s(n+1)} ⊃ F_{n+1}
↑_{αn}  ^{βn} ↘  ↑_{α_{n+1}}
E_{n} ⊂ B_{n}  ⟶^{ψ_{n,n+1}}  B_{n+1} ⊃ E_{n+1} ,

to satisfy the conditions (0.1)-(0.5). Our construction are divided into several steps. In order to provide the reader with a whole picture of the construction, we first give an outline of it. Then the detailed construction will follow.

Outline of the construction. We will construct the following diagram.

[figure: C; ξ_{2}; P_{1}A_{m_{1}}P_{1}; ϕ_{m_{1},s(n+1)}; ϕ_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}ϕ_{m_{1},s(n+1)}(P_{1}); A_{s(n)}; ≈ϕ_{s(n),m_{1}}; θ; P_{0}A_{m_{1}}P_{0}; ξ_{1}; ϕ_{m_{1},m_{2}}; RA_{m_{2}}R; θ_{0}+θ_{1}; λ∘α′; RA_{m_{2}}R; θ_{0}+θ_{1}; (Q_{0}+Q_{1})A_{s(n+1)}(Q_{0}+Q_{1}); α′′; Adu; ξ_{3}; D; ξ_{4}; (Q_{2})A_{s(n+1)}(Q_{2}); ⨁; ⨁; ≈ϕ_{m_{2},s(n+1)}; 1; 2; u; 3; B_{n}; α; B; ψ; α_{n}; β; ⨁]

This large picture consists of several smaller diagrams, each of which is called a sub-diagram. There are two kinds of sub-diagrams. The sub-diagrams of the first kind are labeled by the numbers 1, 2, 3 and the letter u (in the centers of the sub-diagrams). These sub-diagrams are almost commutative in some sense. For example, the one in the center of the large picture, labeled by the letter u consists of two composite maps (θ_{0} + θ_{1}) ∘ (λ ∘ α′) ∘ β and (θ_{0} + θ_{1}) ∘ (ϕ_{m_{1},m_{2}}|_{P_{0}A_{m_{1}} P_{0}}). They are almost equal to each other on a given finite set up to unitary equivalence.

The sub-diagrams of the second kind are those two labeled by “≈ ϕ_{s(n),m_{1}}” and “≈ ϕ_{m_{2},s(n+1)}”. They describe the approximate decompositions of the given maps “ϕ_{s(n),m_{1}}” and “ϕ_{m_{2},s(n+1)}|_{RA_{m_{2}} R}”.

All the maps in the above picture are homomorphisms except β, θ, and θ_{0} + θ_{1} (which are represented by broken line arrows). These maps are linear ∗- contractions which are almost multiplicative on some given finite sets (i.e., on the sets F_{n} ⊂ A_{s(n)}, F := θ(F_{n}) ⊂ P_{0}A_{m_{1}}P_{0}, or a certain (large enough) finite subset F′ ⊂ RA_{m_{2}}R_{2}) to within given small numbers (i.e., ε_{n} or some related small numbers).

The sub-diagrams labeled by the numbers 1, 2, and 3 are approximately commutative on certain given finite sets (i.e., E_{n} ⊂ B_{n}, G := ψ(E_{n}) ∪ β(F) ⊂ B (F is from the above paragraph)) to within a small number (i.e., ε_{n}). The subdiagram labeled by the letter u is approximately commutative on a finite set (F := θ(F_{n})) to within a given small number (9ε_{n}) up to unitary equivalence.

The sub-diagrams labeled by “≈ ϕ_{s(n),m_{1}}” and “≈ ϕ_{m_{2},s(n+1)}” are approximate decompositions of ϕ_{s(n),m_{1}} and ϕ_{m_{2},s(n+1)}|_{RA_{m_{2}} R}, respectively. (E.g., the direct sum θ ⊕ (ξ_{2} ∘ ξ_{1}) of the two maps θ and ξ_{2} ∘ ξ_{1} is close to ϕ_{s(n),m_{1}} to within a small number ε_{n} on a given finite set F_{n}.)

The above decomposition of ϕ_{s(n),m_{1}} and the almost commutative sub-diagram labeled by the number 1 are obtained in Step 1 in the detailed proof, applying Theorem 5.32b to A_{s(n)} and α_{n} : B_{n} → A_{s(n)} (and to the finite sets E_{n} and F_{n}). The main purpose of this step is to make the set θ(F_{n}) := F weakly approximately constant to within ε_{n} (the other part ξ_{2}∘ξ_{1} of the decomposition factors through an interval algebra C), which will be useful later when we apply Theorem 1.6.9. (If one assumes in the beginning that the set F_{n} is weakly approximately constant to within ε_{n}, then he does not need this step.)

The sub-diagrams labeled by 2 or u will be explained by another picture later. The almost commutative sub-diagram labeled by the number 3 and the decomposition of ϕ_{m_{2},s(n+1)}|_{RA_{m_{2}} R} (i.e., “≈ ϕ_{m_{2},s(n+1)}” in the picture), are obtained in Step 4, applying Theorem 5.32a to RA_{m_{2}}R and λ ∘ α′ : B → RA_{m_{2}}R (and certain finite subsets of B and RA_{m_{2}}R). The purpose of applying Theorem 5.32a is to construct the map θ_{0} + θ_{1} to satisfy the condition in Theorem 1.6.9 for the two homotopic homomorphisms λ ∘ (ϕ ⊕ r) and ϕ_{m_{1},m_{2}}|_{P_{0}A_{m_{1}} P_{0}} in the next picture, and therefore to obtain the almost commutative sub-diagram up to unitary equivalence—the sub-diagram labeled by u—, (the other part ξ_{4} ∘ ξ_{3} of the decomposition factors through an interval algebra D).

In order to get the parts of the sub-diagram labeled by 2 and u, we need to start with α : B_{n} → P_{0}A_{m_{1}}P_{0}. We describe it in the next picture.

[figure: RA_{m_{2}}R; ϕ_{m_{1},m_{2}}; λ; A(= P_{0}A_{m_{1}}P_{0}); ϕ ⊕ r; h; 4; 2; β; M_{L}(A) ⊕ r(A); B_{n}; B; α; α′; ψ; θ_{0}+θ_{1}; (Q_{0}+Q_{1})A_{s(n+1)}(Q_{0}+Q_{1}); Adu]

By Corollary 1.6.29 (see 1.6.31 also), applied to the homomorphism α from the first picture, we obtain the almost commutative sub-diagrams labeled by the numbers 2 and 4. Then we apply Lemma 1.6.30 to obtain the sub-diagram labeled by the letter h which commutes up to homotopy equivalence. By Theorem 1.6.9 and the property of the map θ_{0} + θ_{1} (from Theorem 5.32a), this sub-diagram leads to the sub-diagram labeled by u in the first picture. With the first picture in mind, we define

B_{n+1} = C ⊕ B ⊕ D,
ψ_{n,n+1} = (ξ_{1} ∘ α_{n}) ⊕ ψ ⊕ (ξ_{3} ∘ ϕ_{m_{1},m_{2}}|_{P_{0}A_{m_{1}} P_{0}} ∘ α),
β_{n} = ξ_{1} ⊕ (β ∘ θ) ⊕ (ξ_{3} ∘ ϕ_{m_{1},m_{2}}|_{P_{0}A_{m_{1}} P_{0}} ∘ θ),

and

α_{n+1} = (ϕ_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}} P_{1}} ∘ ξ_{2}) ⊕ (Adu ∘ α′′) ⊕ ξ_{4}.

In the definitions of ψ_{n,n+1} and α_{n+1}, we use solid line arrows only since these maps are supposed to be homomorphisms (but in the definition of β_{n}, we can use broken line arrows). One can easily verify the conditions (0.1)–(0.5) except that the map ϕ_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}} P_{1}} ∘ ξ_{2} may not automatically satisfy the dichotomy condition (0.5), for which we have to apply Lemma 6.2 to make some modification. Details of the construction. The above outline can be used as a guide to understand the following construction. But the proof below is complete by itself. (We encourage readers to compare the following detailed proof with the two diagrams in the outline.) Among the conditions in the induction assumption, only the dichotomy condition (0.5) of α_{n} is used in the following construction. Step 1. By Theorem 5.32b, applied to α_{n} : B_{n} → A_{s(n)}, E_{n} ⊂ B_{n}, F_{n} ⊂ A_{s(n)}, and ε > 0, there are A_{m_{1}} (m_{1} > s(n)), two orthogonal projections P_{0}, P_{1} ∈ A_{m_{1}} with ϕ_{s(n),m_{1}}(1_{A_{s(n)}}) = P_{0} +P_{1} and P_{0} trivial, a C^{∗}-algebra C — a direct sum of matrix algebras over C[0, 1] or C^{|} —, a unital map θ ∈ Map(A_{s(n)}, P_{0}A_{m_{1}}P_{0})_{1}, a unital homomorphism ξ_{1} ∈ Hom(A_{s(n)}, C)_{1}, an injective unital homomorphism ξ_{2} ∈ Hom(C, P_{1}A_{m_{1}}P_{1})_{1} and a (not necessarily unital) homomorphism α ∈ Hom(B_{n}, P_{0}A_{m_{1}}P_{0}) such that (1.1) ∥ϕ_{s(n),m_{1}}(f) − θ(f) ⊕ (ξ_{2} ∘ ξ_{1})(f)∥ < ε_{n} for all f ∈ F_{n}. (1.2) θ is F_{n}-ε_{n} multiplicative and F := θ(F_{n}) is weakly approximately constant to within ε_{n}. (1.3) ∥α(g) − θ ∘ α_{n}(g)∥ < ε_{n} for all g ∈ E_{n}. (1.4) Both α : B_{n} → P_{0}A_{m_{1}}P_{0} and ξ_{2} : C → P_{1}A_{m_{1}}P_{1} satisfy the dichotomy condition in (0.5). (Thus we finished the construction of the sub-diagrams labeled by the number “1” and “≈ ϕ_{s(n),m_{1}}” of the large diagram in the outline.) Let all the blocks of C be parts of C^{∗}-algebra B_{n+1}. That is,

B_{n+1} = C ⊕ (some other blocks).

The map β_{n} : A_{s(n)} → B_{n+1} and the homomorphism ψ_{n,n+1} : B_{n} → B_{n+1} are defined by

β_{n} = ξ_{1} : A_{s(n)} → C(⊂ B_{n+1}) and ψ_{n,n+1} = ξ_{1} ∘ α_{n} : B_{n} → C ( ⊂ B_{n+1})

for the blocks of C ( ⊂ B_{n+1}). For this part, β_{n} is also a homomorphism. Step 2. Let A = P_{0}A_{m_{1}}P_{0}, F = θ(F_{n}). Since P_{0} is a trivial projection,

A ≅ ⨁ M_{l_{i}}(C(X_{m_{1},i})).

Let rA := ⨁ M_{l_{i}}(C^{|} ) ⊂ A, and r : A → rA be the homomorphism defined by evaluation at certain base points x_{i}^{0} ∈ X_{m_{1},i} (see 1.1.7(h)). Applying Corollary 1.6.29 (see Remark 1.6.31 also) to α : B_{n} → A (notice that α satisfies the dichotomy condition), E_{n} ⊂ B_{n} and F ⊂ A, we obtain the following diagram:

\begin{matrix}A & ⟶^{ϕ⊕r} & M_{L}(A) ⊕ r(A) \\ ↑_{α} & ^{β} ↘ & ↑_{α′} \\ B_{n} & ⟶^{ψ} & B \end{matrix}

such that (2.1) B is a direct sum of matrix algebras over {pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k}, or S^{2}. (2.2) α′ is an injective homomorphism, and β is an F -ε_{n} multiplicative map. (2.3) ϕ : A → M_{L}(A) is a unital simple embedding. r : A → r(A) is the homomorphism defined by evaluations as in 1.1.7(h). (2.4) ∥β ∘ α(g) − ψ(g)∥ < ε_{n} for all g ∈ E_{n}, ∥(ϕ ⊕ r)(f) − α′ ∘ β(f)∥ < ε_{n} for all f ∈ F (:= θ(F_{n})). (Thus we finished the construction of the sub-diagrams labeled by the number “2” and “4” of the second diagram in the outline.) Let all the blocks B be also parts of B_{n+1}, that is,

B_{n+1} = C ⊕ B ⊕ (some other blocks).

The maps β_{n} : A_{s(n)} → B_{n+1}, ψ_{n,n+1} : B_{n} → B_{n+1} are defined by

β_{n} := β ∘ θ : A_{s(n)} ⟶^{θ} A ⟶^{β} B (⊂ B_{n+1})

and

ψ_{n,n+1} := ψ : B_{n} ⟶ B (⊂ B_{n+1})

for the blocks of B(⊂ B_{n+1}). This part of β_{n} is F_{n}-2ε_{n} multiplicative, since θ is F_{n}-ε_{n} multiplicative, β is F -ε_{n} multiplicative, and F = θ(F_{n}). Step 3. By the simplicity of lim(A_{n}, ϕ_{n,m}), for m large enough, the homomorphism ϕ_{m_{1},m}|_{P_{0}A_{m_{1}} P_{0}} is 4M-large in the sense of 1.6.16. By Lemma 1.6.30, applied to ϕ ⊕ r : A → M_{L}(A) ⊕ r(A), there is an A_{m_{2}} and unital homomorphism λ : M_{L}(A) ⊕ r(A) → RA_{m_{2}}R, where R = ϕ_{m_{1},m_{2}}(P_{0}) (write R as ⨁_{j} R^{j} ∈ ⨁_{j} A_{m_{i}}^{j}) such that the diagram

[figure: RA_{m_{2}}R; ϕ_{m_{1},m_{2}}; λ; A(= P_{0}A_{m_{1}}P_{0}); ϕ ⊕ r; M_{L}(A) ⊕ r(A)]

satisfies the following conditions: (3.1) For each block A_{m_{2}}^{j}, the partial map

λ^{·,j} : M_{L}(A) ⊕ r(A) ⟶ R^{j}A_{m_{2}}^{j}R^{j}

is non zero. Furthermore, either it is injective or it has finite dimensional image — depending on whether SP(A_{m_{2}}^{j}) is a single point space. (3.2) λ ∘ (ϕ ⊕ r) is homotopy equivalent to

ϕ′ := ϕ_{m_{1},m_{2}}|_{A}.

(Thus we finished the construction of the sub-diagram labeled by the letter “h” of the second diagram in the outline.) Step 4. Applying Theorem 1.6.9 to the finite set F ⊂ A (which is weakly approximately constant to within ε_{n}), and to two homotopic homomorphisms

ϕ′ and λ ∘ (ϕ ⊕ r) : A ⟶ RA_{m_{2}}R

(with RA_{m_{2}}R in place of C), we obtain a finite set F′ ⊂ RA_{m_{2}}R, δ > 0 and L > 0 as in the Theorem 1.6.9. Let G := ψ(E_{n}) ∪ β(F) ⊂ B. By Theorem 5.32a, applied to RA_{m_{2}}R,

λ ∘ α′ : B ⟶ RA_{m_{2}}R

(which satisfies the dichotomy condition by (2.2) and (3.1)), finite sets G ⊂ B, F′ ⊂ RA_{m_{2}}R, min(ε_{n}, δ) > 0 (in place of ε), and L > 0, there are A_{s(n+1)}, mutually orthogonal projections Q_{0}, Q_{1}, Q_{2} ∈ A_{s(n+1)} with ϕ_{m_{2},s(n+1)}(R) = Q_{0} + Q_{1} + Q_{2}, a C^{∗}-algebra D — a direct sum of matrix algebras over C[0, 1]—, a unital map θ_{0} ∈ Map(RA_{m_{2}}R, Q_{0}A_{s(n+1)}Q_{0}) and four unital homomorphisms θ_{1} ∈ Hom(RA_{m_{2}}R, Q_{1}A_{s(n+1)}Q_{1})_{1}, ξ_{3} ∈ Hom(RA_{m_{2}}R, D)_{1}, ξ_{4} ∈ Hom(D, Q_{2}A_{s(n+1)}Q_{2})_{1}, and α′′ ∈ Hom(B, (Q_{0} + Q_{1})A_{s(n+1)}(Q_{0} + Q_{1}))_{1} such that the following are true: (4.1) ∥ϕ_{m_{2},s(n+1)}(f) − ((θ_{0} + θ_{1}) ⊕ (ξ_{4} ∘ ξ_{3}))(f)∥ < ε_{n} for all f ∈ F′ ⊂ RA_{m_{2}}R. (4.2) ∥α′′(g) − (θ_{0} + θ_{1}) ∘ λ ∘ α′(g)∥ < ε_{n} for all g ∈ G. (4.3) θ_{0} is F′-min(ε_{n}, δ) multiplicative and θ_{1} satisfies that

θ_{1}^{i,j}([q]) > L · [θ_{0}^{i,j}(R^{i})]

for any non zero projection q ∈ R^{i}A_{m_{2}}R^{i}. (4.4) Both α′′ : B ⟶ (Q_{0} + Q_{1})A_{s(n+1)}(Q_{0} + Q_{1}) and ξ_{4} : D → Q_{2}A_{s(n+1)}Q_{2} satisfy the dichotomy condition (0.5). (Thus we finished the construction of the sub-diagrams labeled by the number “3” and “≈ ϕ_{m_{2},s(n+1)}” of the large diagram in the outline. Combined with Step 2 and Step 3, these two sub-diagrams will lead to the sub-diagram labeled by the letter “u” of the large diagram as below.) By the end of 1.1.4, for any blocks A^{i}, A_{m_{2}}^{k} and any non zero projection e ∈ A^{i}, ϕ_{m_{1},m_{2}}^{i,k}(e) ∈ A_{m_{2}}^{k} is a non zero projection. As a consequence of (4.3), we have

[(θ_{1} ∘ ϕ′)(e)] ≥ L · [θ_{0}(R)] (= L · [Q_{0}]),

(Recall that ϕ′ = ϕ_{m_{1},m_{2}}|_{A}). Therefore, θ_{0} and θ_{1} (in place of λ_{0} and λ_{1}) satisfy the condition in Theorem 1.6.9. By Theorem 1.6.9, there is a unitary u ∈ (Q_{0} + Q_{1})A_{s(n+1)}(Q_{0} + Q_{1}) such that

∥(θ_{0} + θ_{1}) ∘ ϕ′(f) − Adu ∘ (θ_{0} + θ_{1}) ∘ λ ∘ (ϕ ⊕ r)(f)∥ < 8ε_{n}, ∀f ∈ F.

Combining it with the second inequality of (2.4), we have (4.5) ∥(θ_{0} + θ_{1}) ∘ ϕ′(f) − Adu ∘ (θ_{0} + θ_{1}) ∘ λ ∘ α′ ∘ β(f)∥ < 9ε_{n}, ∀f ∈ F. Step 5. Finally, let all the blocks of D be the rest of B_{n+1}. Namely, let

B_{n+1} = C ⊕ B ⊕ D,

where C is from Step 1, B is from Step 2, and D is from Step 4. We already have the definitions of β_{n} : A_{s(n)} → B_{n+1} and ψ_{n,n+1} : B_{n} → B_{n+1} for those blocks of C ⊕ B ⊂ B_{n+1} (from Step 1 and Step 2). The definitions of β_{n} and ψ_{n,n+1} for blocks of D, and the homomorphism α_{n+1} : C ⊕ B ⊕ D → A_{s(n+1)} will be given below. The part of β_{n} : A_{s(n)} → D (⊂ B_{n+1}) is defined by

β_{n} = ξ_{3} ∘ ϕ′ ∘ θ : A_{s(n)} ⟶^{θ} A ⟶^{ϕ′} RA_{m_{2}}R ⟶^{ξ_{3}} D.

(Recall that A = P_{0}A_{m_{1}}P_{0} and ϕ′ = ϕ_{m_{1},m_{2}}|_{A}.) Since θ is F_{n}-ε_{n} multiplicative, and ϕ′ and ξ_{3} are homomorphisms, we know that this part of β_{n} is F_{n}-ε_{n} multiplicative. The part of ψ_{n,n+1} : B_{n} → D (⊂ B_{n+1}) is defined by

ψ_{n,n+1} = ξ_{3} ∘ ϕ′ ∘ α : B_{n} ⟶^{α} A ⟶^{ϕ′} RA_{m_{2}}R ⟶^{ξ_{3}} D

which is a homomorphism. The homomorphism α_{n+1} : C ⊕ B ⊕ D → A_{s(n+1)} is defined as follows. Consider the composition

ϕ′′ ∘ ξ_{2} : C ⟶^{ξ_{2}} P_{1}A_{m_{1}}P_{1} ⟶^{ϕ′′} ϕ_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}ϕ_{m_{1},s(n+1)}(P_{1}),

where P_{1} and ξ_{2} are from Step 1, ϕ′′ = ϕ_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}} P_{1}}. Using the dichotomy condition of ξ_{2}, by Lemma 6.2, there is a homomorphism τ : C → ϕ_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}ϕ_{m_{1},s(n+1)}(P_{1}) such that (5.1) ∥τ(f) − (ϕ′′ ∘ ξ_{2})(f)∥ < ε_{n}, ∀f ∈ ξ_{1}(F_{n}) ⊂ C, and (5.2) τ satisfies the dichotomy condition (0.5). Define

α_{n+1}|_{C} = τ : C → ϕ_{m_{1},s(n+1)}(P_{1})A_{s(n+1)}ϕ_{m_{1},s(n+1)}(P_{1}),
[figure: α_{n+1}|_{B} = Adu ∘ α′′ : B ⟶^{α′′} (Q_{0} + Q_{1})A_{s(n+1)}(Q_{0} + Q_{1}); Adu ,]

where α′′ is from Step 4, and define

α_{n+1}|_{D} = ξ_{4} : D ⟶ Q_{2}A_{s(n+1)}Q_{2}.

Finally, choose {a_{n+1 j}}_{j=1}^{∞} ⊂ A_{s(n+1)} and {b_{n+1 j}}_{j=1}^{∞} ⊂ B_{n+1} to be countable dense subsets of unit balls of A_{s(n+1)} and B_{n+1}, respectively. And choose

F_{n+1} = ϕ_{s(n),s(n+1)}(F_{n}) ∪ α_{n+1}(E_{n+1}) ∪ ⋃_{i=1}^{n+1} ϕ_{s(i),s(n+1)}({a_{i1}, a_{i2}, ⋯, a_{i n+1}})

and

E_{n+1} = ψ_{n,n+1}(E_{n}) ∪ β_{n}(F_{n}) ∪ ⋃_{i=1}^{n+1} ψ_{i,n+1}({b_{i1}, b_{i2}, ⋯, b_{i n+1}}).

Thus we obtain the following diagram:

F_{n} ⊂ A_{s(n)} ⟶^{ϕ_{s(n),s(n+1)}} A_{s(n+1)} ⊃ F_{n+1}
↑_{αn}  ^{βn} ↘  ↑_{α_{n+1}}
E_{n} ⊂ B_{n}  ⟶^{ψ_{n,n+1}}  B_{n+1} ⊃ E_{n+1} .

Step 6. Now we need to verify all the conditions (0.1)–(0.5) for the above diagram.

(0.1)–(0.2) hold from the construction (see the constructions of B, C, D in Step 1, Step 2 and Step 4, and E_{n+1}, F_{n+1} in the end of Step 5.) (0.3) follows from the end of Step 1, the end of Step 2, and the part of the definition of β_{n} for D from Step 5. (0.5) follows from (4.4) and (5.2). So we only need to verify (0.4). Combining (1.1) with (4.1), we have

\begin{matrix}∥ϕ_{s(n),s(n+1)}(f) − [(ϕ′′ ∘ ξ_{2} ∘ ξ_{1}) ⊕ ((θ_{0} + θ_{1}) ∘ ϕ′ ∘ θ) ⊕ (ξ_{4} ∘ ξ_{3} ∘ ϕ′ ∘ θ)](f)∥ \\ < ε_{n} + ε_{n} = 2ε_{n} \end{matrix}

for all f ∈ F_{n} (recall that ϕ′′ = ϕ_{m_{1},s(n+1)}|_{P_{1}A_{m_{1}} P_{1}}, ϕ′ := ϕ_{m_{1},m_{2}}|_{P_{0}A_{m_{1}} P_{0}}). Combined with (4.2), (4.5), (5.1) and the definitions of β_{n} and α_{n+1}, the proceeding inequality yields

∥ϕ_{s(n),s(n+1)}(f) − (α_{n+1} ∘ β_{n})(f)∥ < 9ε_{n} + ε_{n} + 2ε_{n} = 12ε_{n} ∀f ∈ F_{n}.

Combining (1.3), the first inequality of (2.4), and the definitions of β_{n} and ψ_{n,n+1}, we have

∥ψ_{n,n+1}(g) − β_{n} ∘ α_{n}(g)∥ < ε_{n} + ε_{n} = 2ε_{n} ∀g ∈ E_{n}.

So we obtain (0.4). The theorem follows from Proposition 6.1.

⊓ ⊔

Remark 6.4. In the proof of the above theorem, if there is at least one block of B_{n+1} having spectrum of forms S^{1}, T_{II,k}, T_{III,k}, or S^{2}, then we can chose the map ψ_{n,n+1} to be injective (e.g., the map ψ in Step 2 can be chosen to be injective). Hence, in general, we can make the maps ψ_{n,m}, in the inductive system (B_{n}, ψ_{n,m}), injective. (Note that if no space of S^{1}, T_{II,k}, T_{III,k} or S^{2} appears, then it is easy to make the maps injective; see Theorem 2.2.1 of [Li2]).

Remark 6.5. By Lemma 1.3.3, our main result Theorem 6.3 also holds for general simple AH inductive limit C^{∗}-algebras lim(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}) with uniformly bounded dimensions of X_{n,i}, where P_{n,i} ∈ M_{[n,i]}(C(X_{n,i})) are projections. That is, such an AH algebra can be written as an inductive limit of a system (B_{n} = ⨁_{i=1}^{s_{n}} Q_{n,i}M_{{n,i}}(C(Y_{n,i}))Q_{n,i}, ψ_{n,m}), where Y_{n,i} are the spaces: {pt}, [0, 1], S^{1}, T_{II,k}, T_{III,k} and S^{2}, and Q_{n,i} ∈ M_{{n,i}}(C(Y_{n,i})) are projections.

6.6. Suppose that a simple C^{∗}-algebra A is an inductive limit of matrix algebras over X_{n,i}, where X_{n,i} are the spaces of forms {pt}, [0, 1], S^{1}, S^{2}, T_{II,k} or T_{III,k}. Suppose that K_{∗}(A) is torsion free. Then it can be proved that for each fixed algebra A_{n}, integer N > 0, there is an A_{m} such that

\frac{rankϕ_{n,m}^{i,j}(1_{A_{n}^{i}})}{rank(1_{A_{n}^{i})}} ≥ N,

Documenta Mathematica 7 (2002) 255–461

and that (ϕ_{n,m})_{∗}(torK_{∗}(A_{n})) = 0. Based on this, using the argument from §4 of [G2], we know that for any F ⊂ A_{n}, ε > 0, if N is large enough, then the above ϕ_{n,m} is homotopic to a homomorphism ψ : A_{n} → ϕ_{n,m}(1_{A_{n}})A_{m}ϕ_{n,m}(1_{A_{n}}) satisfying ψ(F) ⊂_{ε} C, where C is a direct sum of matrix algebras over spaces {pt}, [0, 1] and S^{1}. (See [G1] and the proof of Lemma 5.6 of [EGL] also.) Using the above fact, the following Corollary is a direct consequence of our Main Theorem and its proof. (In fact, since the algebras M_{k}(C^{|} ), M_{k}(C[0, 1]) and M_{k}(C(S^{1})) are stably generated, the proof is much simpler (see §3 of [Li3]).

Corollary 6.7. Suppose that A is a simple C^{∗}-algebra which is an inductive limit of an AH system with uniformly bounded dimensions of local spectra. If K_{∗}(A) is torsion free, then it is an inductive limit of matrix algebras over C(S^{1}).

Combining the above corollary with [El2] (see [NT] also), we have the following theorem.

Theorem 6.8. Suppose that A= lim_{n→∞}(A_{n}= ⨁_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i},ϕ_{n,m})

and B = lim_{n→∞}(B_{n} = ⨁_{i=1}^{s_{n}} Q_{n,i}M_{{n,i}}(C(Y_{n,i}))Q_{n,i}, ψ_{n,m}) are unital simple

inductive limit algebras with uniformly bounded dimensions of local spectra X_{n,i} and Y_{n,i}, respectively. Suppose that K_{∗}(A) = K_{∗}(B) are torsion free. Suppose that there is an isomorphism of ordered groups

ϕ_{0} : K_{0}A ⟶ K_{0}B

taking [1] ∈ K_{0}A into [1] ∈ K_{0}B, that there is a group isomorphism

ϕ_{1} : K_{1}A ⟶ K_{1}B

and that there is an isomorphism between compact convex sets

ϕ_{T} : T B ⟶ T A,

where T A and T B denote the simplices of tracial states of A and B, respectively. Suppose that ϕ_{0} and ϕ_{T} are compatible, in the sense that

τ(ϕ_{0}g) = ϕ_{T} (τ)(g), g ∈ K_{0}A, τ ∈ T B.

It follows that there exists an isomorphism

ϕ : A ⟶ B

giving rise to ϕ_{0}, ϕ_{1}, ϕ_{T}.

Remark 6.9. Since the C^{∗}-algebras C(T_{II,k}), C(T_{III,k}) and C(S^{2}) are not stably generated, our proof heavily depends on the results that, certain G-δ multiplicative maps (with parts of point evaluations of sufficiently large sizes) are approximated by true homomorphisms in §5. We believe that such results should play important role in the future study of general simple C^{∗}-algebras (with or without real rank zero property).

Remark 6.10. From a result of J. Villadsen, [V1], one knows that the restriction on the dimensions of the spaces X_{n,i} can not be removed. In [G5]—an appendix to this article, we will show that the condition of uniformly bounded dimensions of local spectra can be replaced by the condition of very slow dimension growth. The main difficulty for this case is that we can not obtain the homomorphism from B_{n} to A_{s(n)} as the homomorphism α_{n} in the above proof. (The α_{n} in this case will be only a sufficiently multiplicative map.) But we can still construct homomorphisms ψ_{n} : B_{n} → B_{n+1}, if we carefully choose α_{n} and β_{n}. This case does not create essential difficulty, but makes the proof much longer. We refer it to [G5] , a separate appendix to this paper. It could be an improvement if one can replace the very slow dimension growth condition by the slow dimension growth condition. The author believes that the theorem is also true for this case. In fact, if one can prove the corresponding decomposition results (see Section 4) for the AH-algebras with slow dimension growth, then the Main Theorem in this article would also hold, by the same proof as in [G5].

Corollary 6.11. Suppose that A = lim_{n→∞}(A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})), ϕ_{n,m}) is a simple inductive limit C^{∗}-algebras. Suppose that each of the spaces X_{n,i} is of the forms: {pt}, [0, 1], S^{1}, S^{2}, T_{II,k} or T_{III,k}. And suppose that all the connecting maps ϕ_{n,m} are injective. For any F ⊂ A_{n}, ε > 0, if m is large enough, then there are two mutually orthogonal projections P, Q ∈ A_{m} and two homomorphisms ϕ : A_{n} → P A_{m}P and ψ : A_{n} → QA_{m}Q such that

(1) ∥ϕ_{n,m}(f) − (ϕ ⊕ ψ)(f)∥ < ε for all f ∈ F;

(2) ϕ(F) is weakly approximately constant to within ε and SPV(ϕ) < ε;

(3) ψ factors through matrix algebras over C[0, 1]. Furthermore, if for some i, j, the partial map ϕ_{n,m}^{i,j} : A_{n}^{i} → A_{m}^{j} is homotopic to a homomorphism with finite dimensional image, then the part ϕ of the decomposition ϕ ⊕ ψ corresponding to this partial map can be chosen to be zero (or, equivalently, ϕ_{n,m}^{i,j} itself is close to a homomorphism factoring through a matrix algebra over C[0, 1]).

Proof: It follows from the corollary of 2.3 of [Su] that for any M_{l}(C(X)), ε > 0, there are ε_{1} > 0 and a finite subset F of self adjoint elements of M_{l}(C(X)) (i.e., F ⊂ (M_{l}(C(X)))_{s.a}) such that for any homomorphism ϕ : M_{l}(C(X)) → M_{l_{1}}(C(Y)), if ϕ(F) is weakly approximately constant to within ε_{1}, then SPV(ϕ) < ε. Therefore, for the desired condition (2) above, we only need to make ϕ(F) weakly approximately constant to within min(ε, ε_{1}).

To simplify the notation, we still denote min(ε, ε_{1}) by ε. Now, the main body of the corollary follows from Lemma 6.2 and Theorem 5.32b. Namely, first apply Lemma 6.2 to id : A_{n} → A_{n} (in place of ϕ) and A_{n} in place of both B and A_{n} to find A_{n_{1}} (in place of A_{m}) and homomorphism α : A_{n} → A_{n_{1}} such that α satisfies the dichotomy condition and such that α is sufficiently close to ϕ_{n,n_{1}} on the finite set F. Then apply Lemma 5.32b to A_{n} and F ⊂ A_{n} (in place of B and G ⊂ B), A_{n_{1}} and ϕ_{n,n_{1}}(F) ⊂ A_{n_{1}} (in place of A_{n} and F ⊂ A), and α : A_{n} → A_{n_{1}} (in place of α : B → A_{n}) to construct the desired decomposition. (Note that we use the following trivial fact: If two maps ϕ_{1}, ϕ_{2} : A_{n} → A_{m} are approximately equal to each other to within ε_{1} on the finite set F and the set ϕ_{1}(F) is weakly approximately constant to within ε_{2}, then the set ϕ_{2}(F) is weakly approximately constant to within 2ε_{1} + ε_{2}.) For the last part of the Corollary, one needs to notice the following facts.

(i) In the additional parts of Corollaries 5.22 and 5.23, if the homomorphisms ψ are homomorphisms with finite dimensional images, then the homomorphisms ϕ̃ in the corollaries 5.22 and 5.23 are also homomorphisms with finite dimensional images. (ii) In Lemma 5.28, if both ϕ and ϕ_{1} are homomorphisms factoring through interval algebras (this condition implies that they are homotopic to homomorphisms with finite dimensional images), then the homomorphism ψ in Lemma 5.28 (with [ψ]_{∗} = [ϕ_{2}]_{∗}) can be chosen to be a homomorphism with finite dimensional image. With the above facts, if ϕ_{n,m}^{i,j} is homotopic to a homomorphism with finite dimensional image and if X_{n,i} ≠ S^{1}, then the corresponding part of ϕ in our corollary could be chosen to be a homomorphism with finite dimensional image, and therefore it can also factor through matrix algebras over C[0, 1]. So, we can put it together with the part ψ and hence the part ϕ disappears from the decomposition of this partial map. This proves the additional part for the case X_{n,i} ≠ S^{1}. For the case that X_{n,i} = S^{1}, the additional part of the corollary follows from the following claim. Claim: For any unitary u ∈ A_{n}^{i} and any ε > 0, there is an integer N > n such that if m > N, and if ϕ_{n,m}^{i,j}(u) is in the path connected component of the unit in the unitary group of ϕ_{n,m}^{i,j}(1_{A_{n}^{i}})A_{m}^{j}ϕ_{n,m}^{i,j}(1_{A_{n}^{i}}), then there is a self adjoint element a ∈ ϕ_{n,m}^{i,j}(1_{A_{n}^{i}})A_{m}^{j}ϕ_{n,m}^{i,j}(1_{A_{n}^{i}}) such that

∥ϕ_{n,m}^{i,j}(u) − e^{2πia}∥ < ε.

(Obviously, if ϕ_{n,m}^{i,j} is homotopic to a homomorphism with finite dimensional image, then ϕ_{n,m}^{i,j}(u) is in the path connected component of the unit element in the unitary group of ϕ_{n,m}^{i,j}(1_{A_{n}^{i}})A_{m}^{j}ϕ_{n,m}^{i,j}(1_{A_{n}^{i}}).) The proof of the above claim is exactly the same as the proof of the main theorem of [Phi3]: the simple inductive limit C^{∗}-algebra in our corollary has exponential rank at most 1 + ε. We omit the details.

We point out that, in [EGL], we will only need this result for the case X_{m,j} = S^{2}. Since dim(S^{2}) ≤ 2, P M_{•}(C(S^{2}))P has exponential rank at most 1 + ε. Therefore, the claim for the case X_{m,j} = S^{2} (X_{n,i} = S^{1}) is trivial.

⊓ ⊔

By Lemma 1.3.3, the above corollary also holds for the case of A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, instead of A_{n} = ⨁_{i=1}^{t_{n}} M_{[n,i]}(C(X_{n,i})).

Corollary 6.12. Suppose that A = lim_{n→∞}(A_{n} = ⨁_{i=1}^{t_{n}} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m}) is a simple inductive limit C^{∗}-algebra. Suppose that each of the spaces X_{n,i} is of the forms: {pt}, [0, 1], S^{1}, S^{2}, T_{II,k} or T_{III,k}. And suppose that all the connecting maps ϕ_{n,m} are injective. For any F ⊂ A_{n}, ε > 0, if m is large enough, then there are two mutually orthogonal projections P, Q ∈ A_{m} and two homomorphisms ϕ : A_{n} → P A_{m}P and ψ : A_{n} → QA_{m}Q such that

(1) ∥ϕ_{n,m}(f) − (ϕ ⊕ ψ)(f)∥ < ε for all f ∈ F;

(2) ϕ(F) is weakly approximately constant to within ε and SPV(ϕ) < ε;

(3) ψ factors through matrix algebras over C[0, 1]. Furthermore, if for some i, j, the partial map ϕ_{n,m}^{i,j} : A_{n}^{i} → A_{m}^{j} is homotopic to a homomorphism with finite dimensional image, then the part ϕ of the decomposition ϕ ⊕ ψ corresponding to this partial map can be chosen to be zero (or, equivalently, ϕ_{n,m}^{i,j} itself is close to a homomorphism factoring through a matrix algebra over C[0, 1]).

Proof: By Lemma 1.3.3, there is an inductive system

Ã = lim_{n→∞}(Ã_{n} = ⨁_{i=1}^{t_{n}} M_{{n,i}}(C(X_{n,i})), ϕ̃_{n,m})

such that each P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i} is a corner of M_{{n,i}}(C(X_{n,i})) and ϕ_{n,m} = ϕ̃_{n,m}|_{P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}}. Ã is simple since it is stably isomorphic to a simple C^{∗}-algebra A. ϕ̃_{n,m} are injective since ϕ_{n,m} are injective. Apply Corollary 6.11 to F ∪ \{1_{A_{n}^{i}}\}_{i=1}^{t_{n}} ⊂ A_{n} ⊂ Ã_{n} and \frac{ε}{4} > 0 to obtain ϕ̃ and ψ̃ as the homomorphisms ϕ and ψ in Corollary 6.11. Since

∥(ϕ̃ + ψ̃)(1_{A_{n}^{i}}) − ϕ̃_{n,m}(1_{A_{n}^{i}})∥ < \frac{ε}{4}, ∀i,

there is a unitary u ∈ Ã_{m} such that ∥u − 1∥ < \frac{ε}{2} and

u((ϕ̃ + ψ̃)(1_{A_{n}^{i}}))u^{∗} = ϕ̃_{n,m}(1_{A_{n}^{i}}) = ϕ_{n,m}(1_{A_{n}^{i}}), ∀i

Finally, let

ϕ = (Adu ∘ ϕ̃)|_{A_{n}} and ψ = (Adu ∘ ψ̃)|_{A_{n}}

to obtain our corollary.

⊓ ⊔

References

[Bl1] B. Blackadar, Matricial and ultra-matricial topology, Operator Algebras, Mathematical Physics, and Low Dimensional Topology (R. H. Herman and B. Tanbay, eds.), A K Peters, Massachusetts, 1993, pp. 11–38.

[Bl2] B. Blackadar, K-Theory for Operator Algebras, Springer-Verlag, New York/Berlin/Heidelberg, 1986.

[Bl3] B. Blackadar, Symmetries of the CAR Algebras, Ann. of Math., 131(1990), 589–623.

[BDR] B. Blackadar, M. Dadarlat, and M. Rørdam, M. The real rank of inductive limit C*-algebras, Math. Scand. 69 (1991), 211–216.

[Br] O. Bratteli, Inductive limits of finite dimensional C^{∗}-algebras, Trans. A.M.S., 171(1972), 195–234.

[Ch] E. Christensen, Near inclusions of C^{∗}-algebras, Acta Math. 144(1980) 249-265.

[Con] A. Connes, Noncommutative Geometry, Academic press, New York, Tokyo, 1995.

[Cu] J. Cuntz, K-theory for certain C^{∗}-algebras, Ann. Math. 113 (1981), 181-197.

[D1] M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C^{∗}-algebras, K-theory 9 (1995), 117–137.

[D2] M. Dadarlat, Reduction to dimension three of local spectra of real rank zero C^{∗}-algebras, J. Reine Angew. Math. 460 (1995), 189–212.

[DG] M. Dadarlat and G. Gong, A classification result for approximately homogeneous C^{∗}-algebras of real rank zero , Geometric and Functional Analysis, 7(1997) 646–711.

[DNNP] M. Dadarlat, G. Nagy, A. Nemethi, and C. Pasnicu, Reduction of topological stable rank in inductive limits of C*-algebras, Pacific J. Math. 153 (1992), 267–276.

[DN] M. Dadarlat and Nemethi, Shape theory and (connective) K-theory, J. Operator Theory 23 (1990), 207–291.

[Da] K. Davidson, C^{∗}-algebras by examples, Fields Institute Monographs, 6, A.M.S. Providence, R.I.

[Ell1] G. A. Elliott, A classification of certain simple C^{∗}-algebras, Quantum and Non-Commutative Analysis (editors, H. Araki et al.), Kluwer, Dordrecht, 1993, pp. 373–385.

[Ell2] G. A. Elliott, A classification of certain simple C^{∗}-algebras, II, J. Ramanujan Math. Soc. 12 (1997), 97–134.

[Ell3] G. A. Elliott, The classification problem for amenable C^{∗}-algebras, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland, 1994 (editors, S.D. Chattrji), Birkhäuser, Basel, 1995, pp. 922–932.

[Ell4] Elliott, G.A., On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra 38 (1976), 29–44.

[Ell5] Elliott, G.A., On the classification of C*-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179–219.

[EE] G. A. Elliott and D. E. Evans, The structure of irrational rotation C^{∗}-algebras, Ann. of Math. 138 (1993), 477–501.

[EG1] G. A. Elliott and G. Gong, On inductive limits of matrix algebras over two-tori, Amer. J. Math. 118 (1996), 263–290.

[EG2] G. A. Elliott and G. Gong, On the classification of C^{∗}-algebras of real rank zero, II, Ann. of Math. 144 (1996), 497–610.

[EGJS] G. A. Elliott, G. Gong, X. Jiang, and H. Su, A classification of simple limits of dimension drop C^{∗}-algebras, Fields Institute Communications 13 (1997), 125–143.

[EGL] G. A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit C^{∗}-algebras, II : The isomorphism theorem, preprint.

[EGLP] G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Abelian C^{∗}- subalgebras of C^{∗}-algebras of real rank zero and inductive limit C^{∗}- algebras, Duke Math. J. 83 (1996), 511–554.

[G1] G. Gong, Approximation by dimension drop C^{∗}-algebras and classification, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), 40–44.

[G2] G. Gong, Classification of C^{∗}-algebras of real rank zero and unsuspended E-equivalence types, J. Funct. Anal. 152(1998), 281–329.

[G3] G. Gong, On inductive limits of matrix algebras over higher dimensional spaces, Part I, Math. Scand. 80 (1997), 45–60.

[G4] G. Gong, On inductive limits of matrix algebras over higher dimensional spaces, Part II, Math. Scand. 80(1997), 61–100.

[G5] G. Gong, Simple inductive limit C^{∗}-algebras with very slow dimension growth: An appendix for “On the classification of simple inductive limit C^{∗}-algebras, I: The reduction theorem”, Preprint.

[GL1] G. Gong and H. Lin, The exponential rank of inductive limit C^{∗}- algebras,, Math. Scand. 71 (1992), 301–319.

[GL2] G. Gong and H. Lin, Almost multiplicative morphisms and K-theory, International J. of Math. 11(2000) 983-1000.

[HV] P. Halmos and H. Vaughan, Marriage problems, Amer. J. of Math. 72 (1950), 214–215.

[Hu] D. Husemoller, Fibre Bundles, McGraw–Hill, New York, 1966; reprinted in Springer-Verlag Graduate Texts in Mathematics.

[JS1] X. Jiang and H. Su, On a simple unital projectionless C^{∗}-algebra, American J. of Math. 121(1999), 359–413.

[JS2] X. Jiang and H. Su, A classification of simple inductive limits of splitting interval algebras, J. Funct. Anal., 151(1997), 50–76.

[Kir] E. Kirchberg, The classification of purely infinite C^{∗}-algebras using Kasparov’s theory, preprint.

[Li1] L. Li, On the classification of simple C^{∗}-algebras: Inductive limits of matrix algebras over trees, Mem. Amer. Math. Soc., no. 605, vol. 127, 1997.

[Li2] L. Li, Simple inductive limit C^{∗}-algebras: Spectra and approximation by interval algebras, J. Reine. Angew Math. 507 (1999), 57–79.

[Li3] L. Li, Classification of simple C^{∗}-algebras: Inductive limits of matrix algebras over 1-dimensional spaces, J. Funct. Anal.192 (2002) 1-51.

[Lo] T. Loring, Lifting solutions to perturbing problems in C^{∗}-algebras, 8, Fields Institute monograph, Providence, R.I.

[NT] K. Nielsen, and K. Thomsen, Limit of circle algebras, Exposition Math 14(1996) 17–56.

[Phi1] N. C. Phillips, A classification theorem for nuclear purely infinite simple C^{∗}-algebras, Doc. Math. 5 (2000), 49-114.

[Phi2] N. C. Phillips, How many exponentials?, American J. of Math. 116 (1994) 1513–1543.

[Phi3] N. C. Phillips, Reduction of exponential rank in direct limits of C^{∗}- algebras, Canad. J. Math. 46 (1994), 818–853.

[R] M. Rørdam, Classification of inductive limits of Cuntz algebras, J. Reine Angew. Math. 440 (1993), 175–200.

[Sch] C. Schochet, Topological methods for C^{∗}-algebras IV: mod p homology, Pacific J. Math. 114 (1984), 447–468.

[Se] G. Segal, K-homology theory in K-theory and operator algebras, Lecture Notes in Mathematics 575, Springer-Verlag, 1977, pp. 113–127.

[St] J. R. Stallings, Lectures on Polyhedral Topology (notes by G. A. Swarup) 1967, Tata Institute of Fundamental Research, Bombay.

[Su] H. Su, On the classification of C^{∗}-algebras of real rank zero: Inductive limits of matrix algebras over non-Hausdorff graphs, Mem. Amer. Math. Soc. no. 547, vol. 114 (1995).

[Th1] K. Thomsen, Limits of certain subhomogeneous C^{∗}-algebras, Mem. Soc. Math. Fr. 71 (1999).

[V1] J. Villadsen, Simple C^{∗}-algebras with perforation, J. Funct. Anal., 154(1998), 110–116.

[V2] J. Villadsen, On stable rank of Simple C^{∗}-algebras, J. of Amer. Math. Soc. 12(1999), 1091–1102.

[Wh] G. Whitehead, Elements of Homotopy theory, Springer-Verlag, 1978.

[Zh] S. Zhang, A property of purely infinite simple C^{∗}-algebras, Proc. Amer. Math. Soc. 109 (1990), 717–720.

Guihua Gong

Department of Mathematics

University of Puerto Rico, Rio Piedras San Juan, PR 00931-3355, USA ggong@goliath.cnnet.clu.edu 462

Documenta Mathematica 7 (2002)