A Classification Theorem

for Nuclear Purely Infinite Simple C^{∗}-Algebras^{1}

N. Christopher Phillips

Received: January 6, 1998 Revised: December 31, 1999

Communicated by Joachim Cuntz

Abstract. Starting from Kirchberg’s theorems announced at the operator algebra conference in Genève in 1994, namely O_{2} ⊗ A ≅ O_{2} for separable unital nuclear simple A and O_{∞} ⊗ A ≅ A for separable unital nuclear purely infinite simple A, we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C^{∗}-algebras. It follows that if A and B are unital separable nuclear purely infinite simple C^{∗}-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_{∗}(A) to K_{∗}(B) which preserves the K_{0}-class of the identity, then A ≅ B.

Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms t ↦ φ_{t} and t ↦ ψ_{t} from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t ↦ u_{t} in the unitization B^{+} such that ∥u_{t}φ_{t}(a)u_{t}^{∗} − ψ_{t}(a)∥ → 0 for all a in A. We prove the following two results on deformations and unitary equivalence. Let A be separable, nuclear, unital, and simple, and let D be unital. Then any asymptotic morphism from A to K ⊗ O_{∞} ⊗ D is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from A to K ⊗ O_{∞} ⊗ D are necessarily asymptotically unitarily equivalent. We also give some nonclassification results for the nonnuclear case.

1991 Mathematics Subject Classification: Primary 46L35; Secondary 19K99, 46L80.

0 Introduction

We prove that the isomorphism class of a separable nuclear unital purely infinite simple C^{∗}-algebra satisfying the Rosenberg-Schochet Universal Coefficient Theorem is completely determined by its K-theory. More precisely, let A and B be separable nuclear unital purely infinite simple C^{∗}-algebras which satisfy the Universal Coefficient Theorem, and suppose that there is a graded isomorphism α : K_{∗}(A) → K_{∗}(B) such that α([1_{A}]) = [1_{B}] in K_{0}(B). Then there is an isomorphism φ : A → B such that φ_{∗} = α. This theorem follows from a result asserting that whenever A and B are separable nuclear unital purely infinite simple C^{∗}-algebras (not necessarily satisfying the Universal Coefficient Theorem) which are KK-equivalent via a class in KK-theory which respects the classes of the identities, then there is an isomorphism from A to B whose class in KK-theory is the given one.

As intermediate results, we prove some striking facts about homomorphisms and asymptotic morphisms from a separable nuclear unital simple C^{∗}- algebra to the tensor product of a unital C^{∗}-algebra and the Cuntz algebra O_{∞}. If A and D are any two C^{∗}-algebras, we say that two homomorphisms φ, ψ : A → D are asymptotically unitarily equivalent if there is a continuous unitary path t ↦ u_{t} in D̃ such that lim_{t→∞} u_{t}φ(a)u_{t}∗ = ψ(a) for all a ∈ A. (Here D̃ = D if D is unital, and D̃ is the unitization D^{+} if D is not unital.) Note that asymptotic unitary equivalence is a slightly strengthened form of approximate unitary equivalence, and is an approximate form of unitary equivalence. Our results show that if A is separable, nuclear, unital, and simple, and D is separable and unital, then KK^{0}(A, D) can be computed as the set of asymptotic unitary equivalence classes of full homomorphisms from A to K ⊗ O_{∞} ⊗ D, with direct sum as the operation. Note that we use something close to unitary equivalence, and that there is no need to use asymptotic morphisms, no need to take suspensions, and (essentially because O_{∞} is purely infinite) no need to form formal differences of classes. We can furthermore replace A by K ⊗ O_{∞} ⊗ A, in which case the Kasparov product reduces exactly to composition of homomorphisms. These results can be thought of as a form of unsuspended E-theory. (Compare with [16], but note that we don’t even need to use asymptotic morphisms.) There are also perturbation results: any asymptotic morphism is in fact asymptotically unitarily equivalent (with a suitable definition) to a homomorphism.

We also present what is now known about how badly the classification fails in the nonnuclear case. There are separable purely infinite simple C^{∗}-algebras A with O_{∞} ⊗A ≅ A̸ (Dykema–Rørdam), there are infinitely many nonisomorphic separable exact purely infinite simple C^{∗}-algebras A with O_{∞} ⊗ A ≅ A and K_{∗}(A) = 0 (easily obtained from results of Haagerup and Cowling–Haagerup), and for given K-theory there are uncountably many nonisomorphic separable nonexact purely infinite simple C^{∗}-algebras with that K-theory.

Nuclear Purely Infinite Simple C^{∗}-Algebras 51

Classification of C^{∗}-algebras started with Elliott’s classification [19] of AF algebras up to isomorphism by their K-theory. It received new impetus with his successful classification of certain C^{∗}-algebras of real rank zero with nontrivial K_{1}-groups. We refer to [21] for a recent comprehensive list of work in this area. The initial step toward classification in the infinite case was taken in [8], and was quickly followed by a number of papers [48], [49], [33], [34], [22], [50], [35], [7], [51], [32], [36]. In July 1994, Kirchberg announced [27] a breakthrough: proofs that if A is a separable nuclear unital purely infinite simple C^{∗}-algebra, then O_{2} ⊗ A ≅ O_{2} and O_{∞} ⊗ A ≅ A. (The proofs, closely following Kirchberg’s original methods, are in [29].) This quickly led to two more papers [44], [52]. Here, we use Kirchberg’s results to nearly solve the classification problem for separable nuclear unital purely infinite simple C^{∗}-algebras; the only difficulty that remains is the Universal Coefficient Theorem. The method is a great generalization of that of [44], in which we replace homomorphisms by asymptotic morphisms and approximate unitary equivalence by asymptotic unitary equivalence. We also need a form of unsuspended E-theory, as alluded to above. The most crucial step is done in Section 2, where we show that, in a particular context, homotopy implies asymptotic unitary equivalence. We suggest reading [44] to understand the basic structure of Section 2.

Kirchberg has in [28] independently derived the same classification theorem we have. His methods are somewhat different, and mostly independent of the proofs in [29]. He proves that homotopy implies a form of unitary equivalence in a different context, and does so by eventually reducing the problem to a theorem of this type in Kasparov’s paper [26]. By contrast, the main machinery in our proof is simply the repeated use of Kirchberg’s earlier results as described above.

This paper is organized as follows. In Section 1, we present some important facts about asymptotic morphisms, and introduce asymptotic unitary equivalence. In Section 2, we prove our main technical results: under suitable conditions, homotopic asymptotic morphisms are asymptotically unitarily equivalent and asymptotic morphisms are asymptotically unitarily equivalent to homomorphisms. These results are given at the end of the section. In Section 3, we prove the basic form (still using asymptotic morphisms) of our version of unsuspended E-theory. Finally, Section 4 contains the classification theorem and some corollaries, as well as the nicest forms of the intermediate results discussed above. It also contains the nonclassification results.

Most of this work was done during a visit to the Fields Institute for Research in Mathematical Sciences during Fall 1994, and I would like to thank the Institute for its support and for the stimulating research environment it provided. I would also like to thank a number of people for useful discussions, either in person or by electronic mail, including Marius Dǎdǎrlat, George Elliott, Uffe Haagerup, Eberhard Kirchberg, Alex Kumjian, Huaxin Lin, Mikael Rørdam, Jonathan Samuel, Claude Schochet, and Shuang Zhang. These discussions have led me to considerable simplification of the arguments and improvement of the terminology. I am also grateful to the referee for a careful reading of the paper and useful suggestions. In addition, the participants of a seminar on an earlier version, including Siegfried Echterhof, Ralf Meyer, Bernhard Neubüser, Christian Valqui, and Wilhelm Winter, made a number of helpful comments.

Throughout this paper, U(D) denotes the unitary group of a unital C^{∗}- algebra D, and U_{0}(D) denotes the connected component of U(D) containing 1. We will use repeatedly and without comment Cuntz’s result that K_{1}(D) = U(D)/U_{0}(D) for a unital purely infinite simple C^{∗}-algebra D, as well as his corresponding result that K_{0}(D) is the set of Murray-von Neumann equivalence classes of nonzero projections [14]. We similarly use Kasparov’s KK-theory [26], and we recall here (and do not mention again) that every separable nonunital purely infinite simple C^{∗}-algebra has the form K ⊗ D for a unital purely infinite simple C^{∗}-algebra D [61].

1 Asymptotic morphisms and asymptotic unitary equivalence

The basic objects we work with in this paper are asymptotic morphisms. In the first subsection, we state for convenient reference some of the facts we need about asymptotic morphisms, and establish notation concerning them. In the second subsection, we define and discuss full asymptotic morphisms; fullness is used as a nontriviality condition later in the paper. In the third subsection, we introduce asymptotic unitary equivalence of asymptotic morphisms. This relation is the appropriate version of unitary equivalence in the context of asymptotic morphisms, and will play a fundamental role in Sections 2 and 3.

1.1 Asymptotic morphisms and asymptotic unitary equivalence

Asymptotic morphisms were introduced by Connes and Higson [11] for the purpose of defining E-theory, a simple construction of KK-theory (at least if the first variable is nuclear). In this subsection, we recall the definition and some of the basic results on asymptotic morphisms, partly to establish our notation and partly for ease of reference. We also prove a few facts that are well known but seem not to have been published. We refer to [11], and the much more detailed paper [54], for the details of the rest of the development of E-theory.

If X is a compact Hausdorff Hausdorff space, then C(X, D) denotes the C^{∗}- algebra of all continuous functions from X to D, while if X is locally compact Hausdorff Hausdorff, then C_{0}(X, D) denotes the C^{∗}-algebra of all continuous functions from X to D which vanish at infinity, and C_{b}(X, D) denote the C^{∗}- algebra of all bounded continuous functions from X to D.

We begin by recalling the definition of an asymptotic morphism.

1.1.1 Definition. Let A and D be C^{∗}-algebras, with A separable. An asymptotic morphism φ : A → D is a family t → φ_{t} of functions from A to D, defined for t ∈ [0, ∞), satisfying the following conditions:

(1) For every a ∈ A, the function t ↦ φ_{t}(a) is continuous from [0, ∞) to D.

(2) For every a, b ∈ A and α, β ∈ C, the limits

lim_{t→∞}(φ_{t}(αa + βb) − αφ_{t}(a) − βφ_{t}(b)),

lim_{t→∞}(φ_{t}(ab) − φ_{t}(a)φ_{t}(b)), and lim_{t→∞}(φ_{t}(a^{∗}) − φ_{t}(a)^{∗})

are all zero.

1.1.2 Definition. ([11]) Let φ and ψ be asymptotic morphisms from A to D.

(1) We say that φ and ψ are asymptotically equal (called “equivalent” in [11]) if for all a ∈ A, we have lim_{t→∞}(φ_{t}(a) − ψ_{t}(a)) = 0.

(2) We say that φ and ψ are homotopic if there is an asymptotic morphism ρ : A → C([0, 1], D) whose restrictions to {0} and {1} are φ and ψ respectively. In this case, we refer to α ↦ ρ^{(α)} = ev_{α} ∘ ρ (where ev_{α} : C([0, 1], D) → D is evaluation at α) as a homotopy from φ to ψ, or as a continuous path of asymptotic morphisms from φ to ψ.

The set of homotopy classes of asymptotic morphisms from A to D is denoted [[A, D]], and the homotopy class of an asymptotic morphism φ is denoted [[φ]].

It is easy to check that asymptotic equality implies homotopy ([54], Remark 1.11).

1.1.3 Definition. Let φ, ψ : A → K⊗D be asymptotic morphisms. The direct sum φ⊕ψ, well defined up to unitary equivalence (via unitaries in M(K ⊗D)), is defined as follows. Choose any isomorphism δ : M_{2}(K) → K, let \overline{δ} : M_{2}(K ⊗ D) → K ⊗ D be the induced map, and define

(φ ⊕ ψ)_{t}(a) = \overline{δ} (( \begin{matrix}φ_{t}(a) & 0 \\ 0 & ψ_{t}(a) \end{matrix} )) .

Note that any two choices for δ are unitarily equivalent (and hence homotopic).

The individual maps φ_{t} of an asymptotic morphism are not assumed bounded or even linear.

1.1.4 Definition. Let φ : A → D be an asymptotic morphism.

(1) We say that φ is completely positive contractive if each φ_{t} is a linear completely positive contraction.

(2) We say that φ is bounded if each φ_{t} is linear and sup_{t} ∥φ_{t}∥ is finite.

(3) We say that φ is selfadjoint if φ_{t}(a^{∗}) = φ_{t}(a)^{∗} for all t and a.

Unless otherwise specified, homotopies of asymptotic morphisms from A to D satisfying one or more of these conditions will be assumed to satisfy the same conditions as asymptotic morphisms from A to C([0, 1], D).

Note that if φ is bounded, then the formula ψ_{t}(a) = \frac{1}{2}(φ_{t}(a) + φ_{t}(a^{∗})^{∗}) defines a selfadjoint bounded asymptotic morphism which is asymptotically equal to φ. We omit the easy verification that ψ is in fact an asymptotic morphism.

1.1.5 Lemma. ([54], Lemma 1.6.) Let A and D be C^{∗}-algebras, with A separable and nuclear. Then every asymptotic morphism from A to D is asymptotically equal to a completely positive contractive asymptotic morphism. Moreover, the obvious map defines a bijection between the sets of homotopy classes of completely positive contractive asymptotic morphisms and arbitrary asymptotic morphisms. (Homotopy classes are as in the convention in Definition 1.1.4.)

1.1.6 Lemma. Let φ : A → D be an asymptotic morphism. Define φ^{+} : A^{+} → D^{+} by φ_{t}(a + λ · 1) = φ_{t}(a) + λ · 1 for a ∈ A and λ ∈ C. Then φ^{+} is an asymptotic morphism from A^{+} to D^{+}, and is completely positive contractive, bounded, or selfadjoint whenever φ is.

The proof of this is straightforward, and is omitted.

The following result is certainly known, but we know of no reference.

1.1.7 Proposition. Let A be a C^{∗}-algebra which is given by exactly stable (in the sense of Loring [37]) generators and relations (G, R), with both G and R finite. Let D be a C^{∗}-algebra. Then any asymptotic morphism from A to D is asymptotically equal to a continuous family of homomorphisms from A to D (parametrized by [0, ∞)). Moreover, if φ^{(0)} and φ^{(1)} are two homotopic asymptotic morphisms from A to D, such that each φ_{t}^{(0)} and each φ_{t}^{(1)} is a homomorphism, then there is a homotopy α ↦ φ^{(α)} which is asymptotically equal to the given homotopy and such that each φ_{t}^{(α)} is a homomorphism.

Note that it follows from Theorem 2.6 of [38] that exact stability of (G, R) depends only on A, not on the specific choices of G and R.

Proof of Proposition 1.1.7: Theorem 2.6 of [38] implies that the algebra A is semiprojective in the sense of Blackadar [4]. (Also see Definition 2.3 of [38].) We will use semiprojectivity instead of exact stability.

We prove the first statement. Let φ : A → D be an asymptotic morphism. Then φ defines in a standard way (see Section 1.2 of [54]) a homomorphism ψ : A → C_{b}([0, ∞), D)/C_{0}([0, ∞), D). Let

I_{n}(D) = {f ∈ C_{b}([0, ∞), D) : f(t) = 0 for t ≥ n}.

Then C_{0}([0, ∞), D) = \overline{⋃_{n=1}^{∞} I_{n}(D)}. Semiprojectivity of A provides an n and a homomorphism σ : A → C_{b}([0, ∞), D)/I_{n}(D) such that the composite of σ and the quotient map

C_{b}([0, ∞), D)/I_{n}(D) → C_{b}([0, ∞), D)/C_{0}([0, ∞), D)

is ψ. Now σ can be viewed as a continuous family of homomorphisms σ_{t} from A to D, parametrized by [n, ∞). Define σ_{t} = σ_{n} for 0 ≤ t ≤ n. This gives the required continuous family of homomorphisms.

The proof of the statement about homotopies is essentially the same. We use C_{b}([0, 1] × [0, ∞), D) in place of C_{b}([0, ∞), D),

J = {f ∈ C_{0}([0, 1] × [0, ∞), D) : f(α, t) = 0 for α = 0, 1}

in place of C_{0}([0, ∞), D), and J ∩ I_{n}([0, 1], D) in place of I_{n}(D). We obtain φ_{t}^{(α)} for all t greater than or equal to some t_{0}, and for all t when α = 0 or 1. We then extend over (0, 1) × [0, t_{0}) via a continuous retraction

[0, 1] × [0, ∞) → ([0, 1] × [t_{0}, ∞)) ∪ ({0} × [0, ∞)) ∪ ({1} × [0, ∞)).

We refer to [11] (and to [54] for more detailed proofs) for the definition of E(A, B) as the abelian group of homotopy classes of asymptotic morphisms from K ⊗SA to K ⊗SB, for the construction of the composition of asymptotic morphisms (well defined up to homotopy), and for the construction of the natural map KK^{0}(A, B) → E(A, B) and the fact that it is an isomorphism if A is nuclear. We do state here for reference the existence of the tensor product of asymptotic morphisms. For the proof, see Section 2.2 of [54].

1.1.8 Proposition. ([11]) Let A_{1}, A_{2}, B_{1}, and B_{2} be separable C^{∗}-algebras, and let φ^{(i)} : A_{i} → B_{i} be asymptotic morphisms. Then there exists an asymptotic morphism ψ : A_{1} ⊗ A_{2} → B_{1} ⊗ B_{2} (maximal tensor products) such that ψ_{t}(a_{1} ⊗ a_{2}) − φ_{t}^{(1)}(a_{1}) ⊗ φ_{t}^{(2)}(a_{2}) → 0 as t → ∞, for all a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. Moreover, ψ is unique up to asymptotic equality.

1.2 Full asymptotic morphisms

In this subsection, we define full asymptotic morphisms. Fullness will be used as a nontriviality condition on asymptotic morphisms in Section 3. It will also be convenient (although not, strictly speaking, necessary) in Section 2.

We make our definitions in terms of projections, because the behavior of asymptotic morphisms on projections can be reasonably well controlled. We do not want to let the asymptotic morphism φ : C_{0}(R) → C_{0}(R), defined by φ_{t}(f) = tf, be considered to be full, since it is asymptotically equal to the zero asymptotic morphism, but in the absence of projections it is not so clear how to rule it out. Fortunately, in the present paper this issue does not arise.

We start with a useful definition and some observations related to the evaluation of asymptotic morphisms on projections.

1.2.1 Definition. Let A and D be C^{∗}-algebras, with A separable. Let p ∈ A be a projection, and let φ : A → D be an asymptotic morphism. A tail projection for φ(p) is a continuous function t ↦ q_{t} from [0, ∞) to the projections in D which, thought of as an asymptotic morphism ψ : C → D via ψ_{t}(λ) = λq_{t}, is asymptotically equal to the asymptotic morphism ψ′_{t}(λ) = λφ_{t}(p).

1.2.2 Remark. (1) Tail projections always exist: Choose a suitable t_{0}, apply functional calculus to \frac{1}{2}(φ_{t}(p) + φ_{t}(p)^{∗}) for t ≥ t_{0}, and take the value at t for t ≤ t_{0} to be the value at t_{0}. (Or use Proposition 1.1.7.)

(2) If φ is an asymptotic morphism from A to D, then a tail projection for φ(p), regarded as an asymptotic morphism from C to D, is a representative of the product homotopy class of φ and the asymptotic morphism from C to A given by p.

(3) A homotopy of tail projections is defined in the obvious way: it is a continuous family of projections (α, t) → q_{t}^{(α)} with given values at α = 0 and α = 1.

(4) If φ is an asymptotic morphism, then it makes sense to say that a tail projection is (or is not) full (that is, generates a full hereditary subalgebra), since fullness depends only on the homotopy class of a projection.

1.2.3 Lemma. Let A and D be C^{∗}-algebras, with A separable. Let φ : A → D be an asymptotic morphism, and let p_{1} and p_{2} be projections in A. If p_{1} is Murray-von Neumann equivalent to a subprojection of p_{2}, then a tail projection for φ(p_{1}) is Murray-von Neumann equivalent to a subprojection of a tail projection for φ(p_{2}).

Proof: Let t ↦ q_{t}^{(1)} and t ↦ q_{t}^{(2)} be tail projections for φ(p_{1}) and φ(p_{2}) respectively. Let v be a partial isometry with v^{∗}v = p_{1} and vv^{∗} ≤ p_{2}. Using asymptotic multiplicativity and the definition of a tail projection, we have

lim_{t→∞} (φ_{t}(v)^{∗}φ_{t}(v) − q_{t}^{(1)}) = 0 and lim_{t→∞} (q_{t}^{(2)}φ_{t}(v)φ_{t}(v)^{∗}q_{t}^{(2)} − q_{t}^{(2)}) = 0.

It follows that for t sufficiently large, q_{t}^{(1)} is Murray-von Neumann equivalent to a subprojection of q_{t}^{(2)}, with the Murray-von Neumann equivalence depending continuously on t. It is easy to extend it from an interval [t_{0}, ∞) to [0, ∞).

1.2.4 Lemma. Let A and D be as in Definition 1.2.1, let α ↦ φ^{(α)} be a homotopy of asymptotic morphisms from A to D, and let p_{0}, p_{1} ∈ A be homotopic projections. Let q^{(0)} and q^{(1)} be tail projections for φ^{(0)}(p_{0}) and φ^{(1)}(p_{1}) respectively. Then q^{(0)} is homotopic to q^{(1)} in the sense of Remark 1.2.1 (3).

Proof: This can be proved directly, but also follows by combining Remark 1.2.2 (2), Proposition 1.1.7, and the fact that products of homotopy classes of asymptotic morphisms are well defined.

1.2.5 Definition. Let A be a separable C^{∗}-algebra which contains a full projection, and let D be any C^{∗}-algebra. Then an asymptotic morphism φ : A → D is full if there is a full projection p ∈ A such that some (equivalently, any) tail projection for φ(p) is full in D.

This definition rejects, not only the identity map of C_{0}(R), but also the identity map of C_{0}(Z). (The algebra C_{0}(Z) has no full projections.) However, it will do for our purposes.

Note that, by Lemma 1.2.3, if a tail projection for φ(p) is full, then so is a tail projection for φ(q) whenever p is Murray-von Neumann equivalent to a subprojection of q.

We now list the relevant properties of full asymptotic morphisms. We omit the proofs; they are mostly either immediate or variations on the proof of Lemma 1.2.3.

1.2.6 Lemma. (1) Fullness of an asymptotic morphism depends only on its homotopy class.

(2) If φ, ψ : A → D are asymptotic morphisms, and if φ is full, then so is the asymptotic morphism φ ⊕ ψ : A → M_{2}(D).

(3) Let B be separable, and have a full projection, and further assume that given two full projections in B, each is Murray-von Neumann equivalent to a subprojection of the other. Then any asymptotic morphism representing the product of full asymptotic morphisms from A to B and from B to D is again full.

The extra assumption in part (3) is annoying, but we don’t see an easy way to avoid it. This suggests that we don’t quite have the right definition. However, in this paper B will almost always have the form K ⊗ O_{∞} ⊗ D with D unital. Lemma 2.1.8 (1) below will ensure that the assumption holds in this case.

1.3 Asymptotic unitary equivalence

Approximately unitarily equivalent homomorphisms have the same class in Rørdam’s KL-theory (Proposition 5.4 of [51]), but need not have the same class in KK-theory. (See Theorem 6.12 of [51], and note that KL(A, B) is in general a proper quotient of KK^{0}(A, B).) Since the theorems we prove in Section 3 give information about KK-theory rather than about Rørdam’s KL-theory, we introduce and use the notion of asymptotic unitary equivalence instead. We give the definition for asymptotic morphisms because we will make extensive technical use of it in this context, but, for reasons to be explained below, it is best suited to homomorphisms.

1.3.1 Definition. Let A and D be C^{∗}-algebras, with A separable. Let φ, ψ : A → D be two asymptotic morphisms. Then φ is asymptotically unitarily equivalent to ψ if there is a continuous family of unitaries t ↦ u_{t} in D̃, defined for t ∈ [0, ∞), such that

lim_{t→∞} ∥u_{t}φ_{t}(a)u_{t}^{∗} − ψ_{t}(a)∥ = 0

for all a ∈ A. We say that two homomorphisms φ, ψ : A → D are asymptotically unitarily equivalent if the corresponding constant asymptotic morphisms with φ_{t} = φ and ψ_{t} = ψ are asymptotically unitarily equivalent.

1.3.2 Lemma. Asymptotic unitary equivalence is the equivalence relation on asymptotic morphisms generated by asymptotic equality and unitary equivalence in the exact sense (that is, u_{t}φ_{t}(a)u_{t}^{∗} = ψ_{t}(a) for all a ∈ A).

Proof: The only point needing any work at all is transitivity of asymptotic unitary equivalence, and this is easy.

1.3.3 Lemma. Let A and D be C^{∗}-algebras, with A separable.

(1) Let φ, ψ : A → K ⊗ D be asymptotically unitarily equivalent asymptotic morphisms. Then φ is homotopic to ψ.

(2) Let φ, ψ : A → K ⊗ D be asymptotically unitarily equivalent homomorphisms. Then φ is homotopic to ψ via a path of homomorphisms.

Proof: (1) Let t ↦ u_{t} ∈ (K ⊗ D)^{+} be an asymptotic unitary equivalence. Modulo the usual isomorphism M_{2}(K) ≅ K, the asymptotic morphisms φ and ψ are homotopic to the asymptotic morphisms φ ⊕ 0 and ψ ⊕ 0 from A to M_{2}(K ⊗ D). Choose a continuous function (α, t) ↦ v_{α,t} from [0, 1] × [0, ∞) to U(M_{2}((K ⊗ D)^{+})) such that v_{0,t} = 1 and v_{1,t} = u_{t} ⊕ u_{t}^{∗} for all t. Define a homotopy of asymptotic morphisms by ρ_{t}^{(α)}(a) = v_{α,t}(φ_{t}(a) ⊕ 0)v_{α,t}^{∗}. Then ρ^{(0)} = φ ⊕ 0 and ρ^{(1)} is asymptotically equal to ψ ⊕ 0. So φ is homotopic to ψ.

(2) Apply the proof of part (1) to the constant paths t ↦ φ and t ↦ ψ. Putting t = 0 gives homotopies of homomorphisms from φ to ρ_{0}^{(1)} and from ψ to ψ ⊕ 0. The remaining piece of our homotopy is taken to be defined for t ∈ [0, ∞], and is given by t ↦ ρ_{t}^{(1)} for t ∈ [0, ∞) and ∞ ↦ ψ ⊕ 0.

1.3.4 Corollary. Two asymptotically unitarily equivalent asymptotic morphisms define the same class in E-theory.

If the domain is nuclear, this corollary shows that asymptotically unitarily equivalent asymptotic morphisms define the same class in KK-theory. Asymptotic unitary equivalence thus rectifies the most important disadvantage of approximate unitary equivalence for homomorphisms. Asymptotic unitary equivalence, however, also has its problems, connected with the extension to asymptotic morphisms. The construction of the product of asymptotic morphisms requires reparametrization of asymptotic morphisms, as in the following definition.

1.3.5 Definition. Let A and D be C^{∗}-algebras, and let φ : A → D be an asymptotic morphism. A reparametrization of φ is an asymptotic morphism from A to D of the form t ↦ φ_{f(t)} for some continuous nondecreasing function f : [0, ∞) → [0, ∞) such that lim_{t→∞} f(t) = ∞.

Other versions are possible: one could replace “nondecreasing” by “strictly increasing”, or omit this condition entirely. The version we give is the most convenient for our purposes.

It is not in general true that an asymptotic morphism is asymptotically unitarily equivalent to its reparametrizations. (Consider, for example, the asymptotic morphism φ : C(S^{1}) → C given by φ_{t}(f) = f(exp(it)).) The product is thus not defined on asymptotic unitary equivalence classes of asymptotic morphisms. (The product is defined on asymptotic unitary equivalence classes when one factor is a homomorphism. We don’t prove this fact because we don’t need it, but see the last part of the proof of Lemma 2.3.5.) In fact, if an asymptotic morphism is asymptotically unitarily equivalent to its reparametrizations, then it is asymptotically unitarily equivalent to a homomorphism, and this will play an important role in our proof. The observation that this is true is due to Kirchberg. It replaces a more complicated argument in the earlier version of this paper, which involved the use throughout of “local asymptotic morphisms”, a generalization of asymptotic morphisms in which there is another parameter. We start the proof with a lemma.

1.3.6 Lemma. Let A and D be C^{∗}-algebras, and let φ : A → D be an asymptotic morphism. Suppose φ is asymptotically unitarily equivalent to all its reparametrizations. Then for any ε > 0 and any finite set F ⊂ A there is M ∈ [0, ∞) such that for any compact interval I ⊂ R and any continuous nondecreasing functions f, g : I → [M, ∞), there is a continuous unitary path t ↦ v_{t} in D̃ satisfying ∥v_{t}φ_{f(t)}(a)v_{t}^{∗} − φ_{g(t)}(a)∥ < ε for all t ∈ I and a ∈ F.

Proof: Suppose the lemma is false. We can obviously change I at will by reparametrizing, so there are ε > 0 and F ⊂ A finite such that for all M ∈ [0, ∞) and all compact intervals I ⊂ R there are continuous nondecreasing functions f, g : I → [M, ∞) for which no continuous unitary path t ↦ v_{t} in D̃ gives ∥v_{t}φ_{f(t)}(a)v_{t}^{∗} − φ_{g(t)}(a)∥ < ε for t ∈ I and a ∈ F. Choose f_{1} and g_{1} for M = M_{1} = 1 and I = I_{1} = [1, 1 + \frac{1}{2}]. Given f_{n} and g_{n}, choose f_{n+1} and g_{n+1} as above for M = M_{n+1} = 1 + max(f_{n}(n + \frac{1}{2}), g_{n}(n + \frac{1}{2})) and I = I_{n+1} = [n + 1, n + 1 + \frac{1}{2}]. By induction, we have M_{n} ≥ n. Let f, g : [0, ∞) → [0, ∞) be the unique continuous functions which are linear on the intervals [n+ \frac{1}{2}, n+1] and satisfy f|_{[n,n+ \frac{1}{2} ]} = f_{n} and g|_{[n,n+ \frac{1}{2} ]} = g_{n}. Since f and g are nondecreasing and satisfy f(t), g(t) ≥ n for t ≥ n, the functions t ↦ φ_{f(t)} and t ↦ φ_{g(t)} are asymptotic morphisms which are reparametrizations of φ. By hypothesis, both are asymptotically unitarily equivalent to φ, and are therefore also asymptotically unitarily equivalent to each other. Let t ↦ v_{t} be a unitary path in D̃ which implements this asymptotic unitary equivalence. Choose T such that for a ∈ F and t > T we have ∥v_{t}φ_{f(t)}(a)v_{t}^{∗} − φ_{g(t)}(a)∥ < ε/2. Restricting to [n, n + \frac{1}{2}] for some n > T gives a contradiction to the choice of M and ε. This proves the lemma.

1.3.7 Proposition. Let A be a separable C^{∗}-algebra, and let φ : A → D be a bounded asymptotic morphism. Suppose that φ is asymptotically unitarily equivalent to all its reparametrizations. Then φ is asymptotically unitarily equivalent to a homomorphism. That is, there exist a homomorphism ω : A → D and a continuous path t ↦ v_{t} of unitaries in D̃ such that for every a ∈ A, we have lim_{t→∞} v_{t}φ_{t}(a)v_{t}^{∗} = ω(a).

Recall from Definition 1.1.4 (2) that bounded asymptotic morphisms are assumed in particular to be linear.

Proof of Proposition 1.3.7: Choose finite sets F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ A whose union is dense in A. Choose a sequence t_{0} < t_{1} < ⋯, with t_{n} → ∞, such that

∥φ_{t}(ab) − φ_{t}(a)φ_{t}(b)∥, ∥φ_{t}(a^{∗}) − φ_{t}(a)^{∗}∥ < 1/2^{n}

for a, b ∈ F_{n} and t ≥ t_{n}, and also such that, as in the previous lemma, for any compact interval I ⊂ R and any continuous nondecreasing functions f, g : I → [t_{n}, ∞), there is a continuous unitary path t ↦ v_{t} in D̃ satisfying ∥v_{t}φ_{f(t)}(a)v_{t}^{∗}− φ_{g(t)}(a)∥ < 2^{−n−1} for all t ∈ I and a ∈ F_{n}. For n ≥ 0 let t ↦ u_{t}^{(n)} be the unitary path associated with the particular choices I = [t_{n}, t_{n+1}], f(t) = t, and g(t) = t_{n}. Set u_{t}^{(n)} = (u_{t_{n}}^{(n)})^{∗}u_{t}^{(n)}. We have ∥(u_{t_{n}}^{(n)})^{∗}φ_{t_{n}}(a)u_{t_{n}}^{(n)} − φ_{t_{n}}(a)∥ < 2^{−n−1} for a ∈ F_{n}, so ∥u_{t}^{(n)}φ_{t}(a)(u_{t}^{(n)})^{∗} − φ_{t_{n}}(a)∥ < 2^{−n} for t ∈ [t_{n}, t_{n+1}] and a ∈ F_{n}. Also note that u_{t_{n}}^{(n)} = 1. Now define a continuous unitary function [0, ∞) → D̃ by

v_{t} = u_{t_{1}}^{(0)} · u_{t_{2}}^{(1)} ⋯ u_{t_{n}}^{(n−1)} · u_{t}^{(n)}

for t_{n} ≤ t ≤ t_{n+1}.

We claim that ω(a) = lim_{t→∞} v_{t}φ_{t}(a)v_{t}^{∗} exists for all a ∈ A. Since sup_{t∈[0,∞)} ∥φ_{t}∥ < ∞, it suffices to check this on the dense subset ⋃_{k=0}^{∞} F_{k}. So let a ∈ F_{k}. We prove that the net t ↦ v_{t}φ_{t}(a)v_{t}^{∗} is Cauchy. Let m ≥ k, and let t ≥ t_{m}. Choose n such that t_{n} ≤ t ≤ t_{n+1}. Then

∥v_{t}φ_{t}(a)v_{t}^{∗} − v_{t_{m}}φ_{t_{m}}(a)v_{t_{m}}^{∗}∥
= ∥[u_{t_{m+1}}^{(m)} · u_{t_{m+2}}^{(m+1)} ⋯ u_{t_{n}}^{(n−1)} · u_{t}^{(n)}]

φ_{t}(a) [u_{t_{m+1}}^{(m)} · u_{t_{m+2}}^{(m+1)} ⋯ u_{t_{n}}^{(n−1)} · u_{t}^{(n)}]^{∗} − φ_{t_{m}}(a)∥

≤ ∥(u_{t}^{(n)}) φ_{t}(a) (u_{t}^{(n)})^{∗} − φ_{t_{n}}(a)∥

+ ∑_{j=m}^{n−1} ∥(u_{t_{j+1}}^{(j)}) φ_{t_{j+1}}(a) (u_{t_{j+1}}^{(j)})^{∗} − φ_{t_{j}}(a)∥

≤ ∑_{j=m}^{n} \frac{1}{2^{j}} < \frac{1}{2^{m−1}} .

Therefore, if r, t ≥ t_{m}, we obtain

∥v_{r}φ_{r}(a)v_{r}^{∗} − v_{t}φ_{t}(a)v_{t}^{∗}∥ < 1/2^{m−2}.

So we have a Cauchy net, which must converge. The claim is now proved.

Since φ_{t} is multiplicative and *-preserving to within 2^{−n} on F_{n} for t ≥ t_{n}, it follows that ω is exactly multiplicative and *-preserving on each F_{n}. Since ∥ω∥ ≤ sup_{t∈[0,∞)} ∥φ_{t}∥ < ∞, it follows that ω is a homomorphism.

In the rest of this section, we prove some useful facts about asymptotic unitary equivalence.

1.3.8 Lemma. Let φ : A → D be an asymptotic morphism, with A unital. Then there is a projection p ∈ D and an asymptotic morphism ψ : A → D which is asymptotically unitarily equivalent to φ and satisfies ψ_{t}(1) = p and ψ_{t}(a) ∈ pDp for all t ∈ [0, ∞) and and a ∈ A.

Proof: Let t ↦ q_{t} be a tail projection for φ(1), as in Definition 1.2.1. Standard results yield a continuous family of unitaries t ↦ u_{t} in D̃ such that u_{0} = 1 and u_{t}q_{t}u_{t}^{∗} = q_{0} for all t ∈ [0, ∞). Define p = q_{0} and define ρ_{t}(a) = u_{t}q_{t}φ_{t}(a)q_{t}u_{t}^{∗} for t ∈ [0, ∞) and a ∈ A. Note that the definition of an asymptotic morphism implies that (t, a) ↦ q_{t}φ_{t}(a)q_{t} is asymptotically equal to φ, and hence is an asymptotic morphism. Thus ρ is an asymptotic morphism which is asymptotically unitarily equivalent to φ.

The only problem is that ρ_{t}(1) might not be equal to p. We do know that ρ_{t}(1) → p as t → ∞. Choose a closed subspace A_{0} of A which is complementary to C · 1, and for a ∈ A_{0} and λ ∈ C define ψ_{t}(a + λ · 1) = ρ_{t}(a) + λp.

1.3.9 Lemma. Let φ, ψ : A → K ⊗ D be asymptotic morphisms, with A and D unital. Suppose that there is a continuous family of unitaries t ↦ u_{t} in the multiplier algebra M(K ⊗ D) such that lim_{t→∞} ∥u_{t}φ_{t}(a)u_{t}^{∗} − ψ_{t}(a)∥ = 0 for all a ∈ A. Then φ is asymptotically unitarily equivalent to ψ.

Proof: We have to show that u_{t} can be replaced by v_{t} ∈ (K ⊗ D)^{+}.

Applying the previous lemma twice, and making the corresponding modifications to the given u_{t}, we may assume that φ_{t}(1) and ψ_{t}(1) are projections p and q not depending on t, and that we always have φ_{t}(a) ∈ pDp and ψ_{t}(a) ∈ qDq.

We now want to reduce to the case p = q. The hypothesis implies that there is t_{0} such that ∥u_{t_{0}}pu_{t_{0}}^{∗} − q∥ < 1/2. Therefore there is a unitary w in (K ⊗ D)^{+} such that wu_{t_{0}}pu_{t_{0}}^{∗}w^{∗} = q. Now if p, q ∈ K ⊗ D are projections which are unitarily equivalent in M(K ⊗ D), then standard arguments show they are unitarily equivalent in (K ⊗ D)^{+}. Therefore conjugating φ by wu_{t_{0}} changes neither its asymptotic unitary equivalence class nor the validity of the hypotheses. We may thus assume without loss of generality that p = q.

Now choose t_{1} such that t ≥ t_{1} implies ∥u_{t}pu_{t}^{∗}−p∥ < 1. Define a continuous family of unitaries by

c_{t} = 1 − p + pu_{t}p(pu_{t}^{∗}pu_{t}p)^{−1/2} ∈ (K ⊗ D)^{+}

for t ≥ t_{1}. (Functional calculus is evaluated in p(K ⊗ D)p.) For any d ∈ p(K ⊗ D)p, we have

∥c_{t}dc_{t}^{∗} − u_{t}du_{t}^{∗}∥ = ∥pdp − c_{t}^{∗}u_{t}pdpu_{t}^{∗}c_{t}∥
≤ 2∥d∥ ∥p − c_{t}^{∗}u_{t}p∥ ≤ 2∥d∥(∥u_{t}p − pu_{t}∥ + ∥p − c_{t}^{∗}pu_{t}p∥).

The first summand in the last factor goes to 0 as t → ∞. Substituting definitions, the second summand becomes ∥p − (pu_{t}^{∗}pu_{t}p)^{1/2}∥, which does the same.

Since φ_{t}(a) ∈ p(K ⊗ D)p for all a ∈ A, and since (using Lemma 1.2 of [54] for the first)

lim sup_{t→∞} ∥φ_{t}(a)∥ ≤ ∥a∥ and lim_{t→∞} ∥u_{t}φ_{t}(a)u_{t}^{∗} − ψ_{t}(a)∥ = 0,

it follows that lim_{t→∞} ∥c_{t}φ_{t}(a)c_{t}^{∗} − ψ_{t}(a)∥ = 0 as well. This is the desired asymptotic unitary equivalence.

2 Asymptotic morphisms to tensor products with O_{∞}.

The purpose of this section is to prove two things about asymptotic morphisms from a separable nuclear unital simple C^{∗}-algebra A to a C^{∗}-algebra of the form K ⊗O_{∞} ⊗D with D unital: homotopy implies asymptotic unitary equivalence, and each such asymptotic morphism is asymptotically unitarily equivalent to a homomorphism. The basic method is the absorption technique used in [35] and [44], and in fact this section is really just the generalization of [44] from homomorphisms and approximate unitary equivalence to asymptotic morphisms and asymptotic unitary equivalence.

There are three subsections. In the first, we collect for reference various known results involving Cuntz algebras (including in particular Kirchberg’s theorems on tensor products) and derive some easy consequences. In the second subsection, we replace approximate unitary equivalence by asymptotic unitary equivalence in the results of [48] and [35]. In the third, we carry out the absorption argument and derive its consequences.

The arguments involving asymptotic unitary equivalence instead of approximate unitary equivalence are sometimes somewhat technical. However, the essential outline of the proof is the same as in the much easier to read paper [44].

2.1 Preliminaries: Cuntz algebras and Kirchberg’s stability theorems

In this subsection, we collect for convenient reference various results related to Cuntz algebras. Besides Rørdam’s results on approximate unitary equivalence and Kirchberg’s basic results on tensor products, we need material on unstable K-theory and hereditary subalgebras of tensor products with O_{∞} and on exact stability of generating relations of Cuntz algebras.

We start with Rørdam’s work [48]; we also use this opportunity to establish our notation. The first definition is used implicitly by Rørdam, and appears explicitly in the work of Ringrose.

We will generally let s_{1}, s_{2}, …, s_{m} be the standard generators of O_{m}, and analogously for O_{∞}.

2.1.1 Definition. ([47], [46]) Let A be a unital C^{∗}-algebra. Then its (C^{∗}) exponential length cel(A) is

sup_{u∈U_{0}(A)} inf {∑_{k=1}^{n} ∥h_{k}∥ : n ∈ N, h_{1}, …, h_{n} ∈ A selfadjoint,

u = exp(ih_{1}) exp(ih_{2}) ⋯ exp(ih_{n})} .

In preparation for the following theorem, and to establish notation, we make the following remark, most of which is in [48], 3.3.

2.1.2 Remark. Let B be a unital C^{∗}-algebra, and let m ≥ 2.

(1) If φ, ψ : O_{m} → B are unital homomorphisms, then the element u = ∑_{j=1}^{m} ψ(s_{j})φ(s_{j})^{∗} is a unitary in B such that uφ(s_{j}) = ψ(s_{j}) for 1 ≤ j ≤ m.

(2) If φ : O_{m} → B is a unital homomorphism, then the formula

λ_{φ}(a) = ∑_{j=1}^{m} φ(s_{j})aφ(s_{j})^{∗}

defines a unital endomorphism λ_{φ} (or just λ when φ is understood) of B.

(3) If φ and λ are as in (2), and if u ∈ B has the form u = vλ(v^{∗}) for some unitary v ∈ B, then vφ(s_{j})v^{∗} = uφ(s_{j}) for 1 ≤ j ≤ m.

2.1.3 Theorem. Let B be a unital C^{∗}-algebra such that cel(B) is finite and such that the canonical map U(B)/U_{0}(B) → K_{1}(B) is an isomorphism. Let m ≥ 2, and let φ, ψ : O_{m} → B, λ : B → B, and u ∈ U(B) be as in Remark 2.1.2 (1) and (2). Then the following are equivalent:

(1) [u] ∈ (m − 1)K_{1}(B).

(2) For every ε > 0 there is v ∈ U(B) such that ∥u − vλ(v^{∗})∥ < ε.

(3) [φ] = [ψ] in KK^{0}(O_{m}, B).

(4) The maps φ and ψ are approximately unitarily equivalent.

Proof: For m even, this is Theorem 3.6 of [48]. In Section 3 of [48], it is also proved that (1) is equivalent to (3) and (2) is equivalent to (4) for arbitrary m, and Theorem 4.2 of [44] implies that (3) is equivalent to (4) for arbitrary m.

We will not actually need to use the equivalence of (3) and (4) for odd m. That cel(D) is finite for purely infinite simple C^{∗}-algebras D was first

proved in [42]. We will, however, apply this theorem to algebras D of the form O_{∞} ⊗ B with B an arbitrary unital C^{∗}-algebra. Such algebras are shown in Lemma 2.1.7 (2) below to have finite exponential length. Actually, to prove the classification theorem, it suffices to know that there is a universal upper bound on cel(C(X) ⊗ B) for B purely infinite and simple. This follows from Theorem 1.2 of [62].

We now state the fundamental results of Kirchberg on which our work depends. These were stated in [27]; proofs appear in [29].

2.1.4 Theorem. ([27]; [29], Theorem 3.8) Let A be a separable nuclear unital simple C^{∗}-algebra. Then O_{2} ⊗ A ≅ O_{2}.

2.1.5 Theorem. ([27]; [29], Theorem 3.15) Let A be a separable nuclear unital purely infinite simple C^{∗}-algebra. Then O_{∞} ⊗ A ≅ A.

We now derive some consequences of Kirchberg’s results.

2.1.6 Corollary. Every separable nuclear unital purely infinite simple C^{∗}- algebra is approximately divisible in the sense of [6].

Proof: It suffices to show that O_{∞} is approximately divisible. Let φ : O_{∞} ⊗ O_{∞} → O_{∞} be an isomorphism, as in the previous theorem. Define ψ : O_{∞} → O_{∞} by ψ(a) = φ(1 ⊗ a). Then ψ is approximately unitarily equivalent to id_{O_{∞}} by Theorem 3.3 of [35]. That is, there are unitaries u_{n} ∈ O_{∞} such that u_{n}φ(1⊗ a)u_{n}^{∗} → a for all a ∈ O_{∞}. Let B ⊂ O_{∞} be a unital copy of M_{2} ⊕ M_{3}. Then for large enough n, the subalgebra u_{n}φ(B ⊗ 1)u_{n}^{∗} of O_{∞} commutes arbitrarily well with any finite subset of O_{∞}.

2.1.7 Lemma. Let D be any unital C^{∗}-algebra. Then:

(1) The canonical map U(O_{∞} ⊗ D)/U_{0}(O_{∞} ⊗ D) → K_{1}(O_{∞} ⊗ D) is an isomorphism.

(2) cel(O_{∞} ⊗ D) ≤ 3π.

Proof: We first prove surjectivity in (1). Let η ∈ K_{1}(O_{∞} ⊗ D). Choose n and u ∈ U(M_{n} ⊗ O_{∞} ⊗ D) such that [u] = η. Let e_{ij} be the standard matrix units in M_{n}. Define (nonunital) homomorphisms

φ : O_{∞} ⊗ D → M_{n} ⊗ O_{∞} ⊗ D and ψ : M_{n} ⊗ O_{∞} ⊗ D → O_{∞} ⊗ D

by

φ(a) = e_{11} ⊗ a and ψ(e_{ij} ⊗ b) = (s_{i} ⊗ 1)b(s_{j}^{∗} ⊗ 1).

Then φ_{∗} is the standard stability isomorphism

K_{1}(O_{∞} ⊗ D) → K_{1}(M_{n} ⊗ O_{∞} ⊗ D).

Also, ψ ∘ φ(a) = (s_{1} ⊗ 1)a(s_{1}^{∗} ⊗ 1) for a ∈ O_{∞} ⊗ D. Since s_{1} ⊗ 1 is an isometry, this implies that ψ ∘ φ is the identity on K-theory. Therefore ψ_{∗} = φ_{∗}^{−1}. Consequently

η = φ_{∗}^{−1}([u]) = ψ_{∗}([u]) = [ψ(u) + 1 − ψ(1)],

showing that η is the class of a unitary in O_{∞} ⊗ D.

Now let u ∈ U(O_{∞} ⊗ D) satisfy [u] = 0 in K_{1}(O_{∞} ⊗ D). We prove that u can be connected to the identity by a path of length at most 3π + ε. This will simultaneously prove (2) and injectivity in (1).

Using approximate divisibility of O_{∞} and approximating u by finite sums of elementary tensors, we can find nontrivial projections e ∈ O_{∞} with ∥u(e ⊗ 1) − (e ⊗ 1)u∥ arbitrarily small. If this norm is small enough, we can find a unitary v ∈ K ⊗ O_{∞} ⊗ D which commutes with e ⊗ 1 and is connected to u by a unitary path of length less than ε/2. Write v = v_{1} + v_{2} with

v_{1} ∈ U(eO_{∞}e ⊗ D) and v_{2} ∈ U((1 − e)O_{∞}(1 − e) ⊗ D).

Choose a partial isometry s ∈ O_{∞} with s^{∗}s = 1 − e and ss^{∗} ≤ e. The proof of Corollary 5 of [42] shows that v can be connected to the unitary

w = v [(e − ss^{∗}) ⊗ 1 + (s ⊗ 1)v_{2}(s ⊗ 1)^{∗} + v_{2}^{∗}]
= 1 − e ⊗ 1 + v_{1} [(s ⊗ 1)v_{2}(s ⊗ 1)^{∗} + (e − ss^{∗}) ⊗ 1]

by a path of length π.

Since O_{∞} is purely infinite, there is an embedding of K ⊗ eO_{∞}e in O_{∞} which extends the obvious identification of e_{11} ⊗ eO_{∞}e with eO_{∞}e. It extends to a unital homomorphism φ : (K ⊗ eO_{∞}e ⊗ D)^{+} → O_{∞} ⊗ D whose range contains w, and such that [φ^{−1}(w)] = 0 in K_{1}(K ⊗eO_{∞}e⊗D). Thus φ^{−1}(w) ∈ U_{0}((K ⊗ eO_{∞}e ⊗ D)^{+}). Theorem 3.8 of [43] shows that the C^{∗} exponential rank of any stable C^{∗}-algebra is at most 2 + ε. An examination of the proof, and of the length of the path used in the proof of Corollary 5 of [42], shows that in fact any stable C^{∗}-algebra has exponential length at most 2π. Thus, in particular, φ^{−1}(w) can be connected to 1 by a unitary path of length 2π + ε/2. It follows that u can be connected to 1 by a unitary path of length at most 3π + ε.

A somewhat more complicated argument shows that in fact cel(O_{∞}⊗D) ≤ 2π. Details will appear elsewhere [45].

2.1.8 Lemma. Let D be a unital C^{∗}-algebra. Then:

(1) Given two full projections in K⊗O_{∞}⊗D, each is Murray-von Neumann equivalent to a subprojection of the other.

(2) If two full projections in K ⊗ O_{∞} ⊗ D have the same K_{0}-class, then they are homotopic.

Proof: Taking direct limits, we reduce to the case that D is separable. Then O_{∞} ⊗ D is approximately divisible by Corollary 2.1.6. It follows from Proposition 3.10 of [6] that two full projections in K ⊗ O_{∞} ⊗ D with the same K_{0}-class are Murray-von Neumann equivalent. Now (2) follows from the fact that Murray-von Neumann equivalence implies homotopy in the stabilization of a unital C^{∗}-algebra.

Part (1) requires slightly more work. Let P be the set of all projections p ∈ O_{∞} ⊗ D such that there are two orthogonal projections q_{1}, q_{2} ≤ p, both Murray-von Neumann equivalent to 1. One readily verifies that P is nonempty and satisfies the conditions (Π_{1})-(Π_{4}) on page 184 of [14]. Therefore, by [14], the group K_{0}(O_{∞} ⊗ D) is exactly the set of Murray-von Neumann equivalence classes of projections in P. Since projections in P are full, Proposition 3.10 of [6] now implies that every full projection is in P. Clearly (1) holds for projections in P. We obtain (1) in general by using the pure infiniteness of O_{∞} to show that every full projection in K ⊗O_{∞} ⊗D is Murray-von Neumann equivalent to a (necessarily full) projection in O_{∞} ⊗ D.

Next, we turn to exact stability. For O_{m}, we need only the following standard result:

2.1.9 Proposition. ([35], Lemma 1.3 (1)) For any integer m, the defining relations for O_{m}, namely s_{j}^{∗}s_{j} = 1 and ∑_{k=1}^{m} s_{k}s_{k}^{∗} = 1 for 1 ≤ j ≤ m, are exactly stable.

We will also need to know about the standard extension E_{m} of O_{m} by the compact operators. Recall from [13] that E_{m} is the universal C^{∗}-algebra on generators t_{1}, …, t_{m} with relations t_{j}^{∗}t_{j} = 1 and (t_{j}t_{j}^{∗})(t_{k}t_{k}^{∗}) = 0 for 1 ≤ j, k ≤ m, j ≠ k. Its properties are summarized in [35], 1.1. In particular, we have lim_{⟶} E_{m} ≅ O_{∞} using the standard inclusions.

Exact stability of the generating relations for E_{m} is known, but we need the following stronger result, which can be thought of as a finite version of exact stability for O_{∞}. Essentially, it says that if elements approximately satisfy the defining relations for E_{m}, then they can be perturbed in a functorial way to exactly satisfy these relations, and that the way the first k elements are perturbed does not depend on the remaining m − k elements.

Recently, Blackadar has proved that in fact O_{∞} is semiprojective in the usual sense [5].

2.1.10 Proposition. For each δ ≥ 0 and m ≥ 2, let E_{m}(δ) be the universal unital C^{∗}-algebra on generators t_{j,δ}^{(m)} for 1 ≤ j ≤ m and relations

∥(t_{j,δ}^{(m)})^{∗}t_{j,δ}^{(m)} − 1∥ ≤ δ and ∥(t_{j,δ}^{(m)}(t_{j,δ}^{(m)})^{∗}) (t_{k,δ}^{(m)}(t_{k,δ}^{(m)})^{∗})∥ ≤ δ

for j ≠ k, and let κ_{δ}^{(m)} : E_{m}(δ) → E_{m} be the homomorphism given by sending t_{j,δ}^{(m)} to the corresponding standard generator t_{j}^{(m)} of E_{m}. Then there are δ(2) ≥ δ(3) ≥ ⋯ > 0, nondecreasing functions f_{m} : [0, δ(m)] → [0, ∞) with lim_{δ→0} f_{m}(δ) = 0 for each m, and homomorphisms φ_{δ}^{(m)} : E_{m} → E_{m}(δ) for 0 ≤ δ ≤ δ(m), satisfying the following properties:

(1) κ_{δ}^{(m)} ∘ φ_{δ}^{(m)} = id_{E_{m}}.

(2) ∥φ_{δ}^{(m)}(t_{j}^{(m)}) − t_{j,δ}^{(m)}∥ ≤ f_{m}(δ).

(3) If 0 ≤ δ ≤ δ′ ≤ δ(m), then the composite of φ_{δ′}^{(m)} with the canonical map from E_{m}(δ′) to E_{m}(δ) is φ_{δ}^{(m)}.

(4) Let ι_{δ}^{(m)} : E_{m}(δ) → E_{m+1}(δ) be the map given by ι_{δ}^{(m)}(t_{j,δ}^{(m)}) = t_{j,δ}^{(m+1)} for 1 ≤ j ≤ m. Then for 0 ≤ δ ≤ δ(m + 1) and 1 ≤ j ≤ m, we have ι_{δ}^{(m)}(φ_{δ}^{(m)}(t_{j}^{(m)})) = φ_{δ}^{(m+1)}(t_{j}^{(m+1)}).

Proof: The proof of exact stability of E_{m}, as sketched in the proof of Lemma 1.3 (2) of [35], is easily seen to yield homomorphisms satisfying the conditions demanded here.

2.1.11 Proposition. Let D be a unital purely infinite simple C^{∗}-algebra. Then any two unital homomorphisms from O_{∞} to D are homotopic. Moreover, if φ, ψ : O_{∞} → D are unital homomorphisms such that φ(s_{j}) = ψ(s_{j}) for 1 ≤ j ≤ m, then there is a homotopy t ↦ ρ_{t} such that ρ_{t}(s_{j}) = φ(s_{j}) for 1 ≤ j ≤ m and all t.

Proof: We prove the second statement; the first is the special case m = 0.

We construct a continuous path t ↦ ρ_{t} of unital homomorphisms from O_{∞} to D, defined for t ∈ [m, ∞) and satisfying the following conditions:

(1) For k ≥ m, for t ∈ [k, ∞), and for 1 ≤ j ≤ k, we have ρ_{t}(s_{j}) = ψ(s_{j}).

(2) ρ_{m} = φ.

Given such a path, ρ_{t}(s_{j}) → ψ(s_{j}) for all j. Since the s_{j} generate O_{∞} as a C^{∗}-algebra, standard arguments show that ρ_{t}(a) → ψ(a) for all a ∈ O_{∞}. We have therefore constructed the required homotopy.

It remains to carry out the construction. We construct the paths t ↦ ρ_{t}, for t ∈ [m, n], by induction on n. The initial step is thus simply to take ρ_{m} = φ. For the induction step, it suffices to do the following. Assume we are given t ↦ ρ_{t}, for t ∈ [m, n], and satisfying ρ_{n}(s_{j}) = ψ(s_{j}) for 1 ≤ j ≤ n. We then extend t ↦ ρ_{t} over t ∈ [n, n + 1] so that ρ_{t}(s_{j}) = ψ(s_{j}) for 1 ≤ j ≤ n and t ∈ [n, n + 1], and in addition ρ_{n+1}(s_{n+1}) = ψ(s_{n+1}).

Let p = ∑_{j=1}^{n} ρ_{n}(s_{j})ρ_{n}(s_{j})^{∗}, which is a projection in D. Then define

e_{0} = ρ_{n}(s_{n+1})ρ_{n}(s_{n+1})^{∗} and e_{1} = ψ(s_{n+1})ψ(s_{n+1})^{∗}.

Both e_{0} and e_{1} are proper projections in the purely infinite simple C^{∗}-algebra (1 − p)D(1 − p) with K_{0}-class equal to [1_{D}], so they are homotopic. It follows that there is a unitary path s ↦ u_{s} in (1 − p)D(1 − p) such that u_{0} = 1 − p and u_{1}e_{0}u_{1}^{∗} = e_{1}. For s ∈ [0, 1/3], define ρ_{n+s}(s_{j}) = ρ_{n}(s_{j}) for 1 ≤ j ≤ n and ρ_{n+s}(s_{j}) = u_{3s}ρ_{n}(s_{j}) for j ≥ n + 1. This yields a homotopy of homomorphisms ρ_{n+s} : O_{∞} → D, with ρ_{n} as already given, and such that the isometries ρ_{n+1/3}(s_{n+1}) and ψ(s_{n+1}) have the same range projection, namely e_{1}, although they themselves are probably not equal.

By a similar argument, we extend the homotopy over [n + 1/3, n + 2/3] in such a way that ρ_{n+s}(s_{j}) is constant for s ∈ [n+1/3, n+2/3] and 1 ≤ j ≤ n+1, and so that ρ_{n+2/3}(s_{n+2}) and ψ(s_{n+2}) also have the same range projection, say f.

Now e_{1} and f are Murray-von Neumann equivalent, so we can identify (e_{1} + f)D(e_{1} + f) with M_{2}(e_{1}De_{1}). Since

w_{1} = ( \begin{matrix}ψ(s_{n+1})ρ_{n+2/3}(s_{n+1})^{∗} & 0 \\ 0 & [ψ(s_{n+1})ρ_{n+2/3}(s_{n+1})^{∗}]^{∗} \end{matrix} )∈ U_{0}(M_{2}(e_{1}De_{1})),

there is a continuous path of unitaries s ↦ w_{s} in M_{2}(e_{1}De_{1}), with w_{0} = 1 and w_{1} as given. For s ∈ [2/3, 1], we now define ρ_{n+s}(s_{j}) = ρ_{n}(s_{j}) for j ≠ n + 1, n + 2, and ρ_{n+s}(s_{j}) = w_{3s−2}ρ_{n+2/3}(s_{j}) for j = n + 1, n + 2. This is again a homotopy, and gives ρ_{n+1}(s_{j}) = ψ(s_{j}) for 1 ≤ j ≤ n + 1, as desired. The induction step is complete.

2.1.12 Corollary. Let D be any unital C^{∗}-algebra, and let p ∈ K ⊗ O_{∞} ⊗ D be a projection. Then O_{∞} ⊗ p(K ⊗ O_{∞} ⊗ D)p ≅ p(K ⊗ O_{∞} ⊗ D)p.

Proof: We may replace p by any Murray-von Neumann equivalent projection. So without loss of generality p ≤ e ⊗ 1 ⊗ 1 for some projection e ∈ K. Using the pure infiniteness of O_{∞}, we can in fact require that e be a rank one projection. That is, we may assume p ∈ O_{∞} ⊗ D.

By Theorem 2.1.5, there is an isomorphism δ : O_{∞} ⊗ O_{∞} → O_{∞}. Using it, we need only consider projections p ∈ O_{∞} ⊗O_{∞} ⊗ D. By the previous proposition and Theorem 2.1.5, a ↦ 1⊗δ(a) is homotopic to id_{O_{∞}⊗O_{∞}}. Therefore such a projection p is homotopic to q = 1 ⊗ (δ ⊗ id_{D})(p), and hence also Murray-von Neumann equivalent to q. Now

q(O_{∞} ⊗ O_{∞} ⊗ D)q ≅ O_{∞} ⊗ [(δ ⊗ id_{D})(p)][O_{∞} ⊗ D][(δ ⊗ id_{D})(p)],

which is unchanged by tensoring with O_{∞} by Theorem 2.1.5.

2.1.13 Corollary. Let D be a unital C^{∗}-algebra. Then the hypotheses on B in Theorem 2.1.3 are satisfied for any unital corner of K ⊗ O_{∞} ⊗ D.

Proof: Combine the previous corollary and Lemma 2.1.7.

2.2 Asymptotic unitary equivalence of homomorphisms from Cuntz algebras

In this subsection, we strengthen the main technical theorems of [48] (restated here as Theorem 2.1.3) and of [35], replacing approximate unitary equivalence by asymptotic unitary equivalence in the conclusions. We use the strong versions to obtain variants of several other known results in which we replace sequences of homomorphisms by continuous paths.

The first lemma contains the essential point in the strengthening of Theorem 2.1.3. Its proof uses the original theorem in a sort of bootstrap argument. The remaining results lead up to the strengthening of the main theorem of [35]. They are proved by modifying the proofs there.

The first two lemmas are stated together, because the proofs are very similar.

2.2.1 Lemma. (Compare with Theorem 2.1.3.) Let D_{0} be a unital C^{∗}-algebra, and let D = O_{∞} ⊗ D_{0}. Let m ≥ 2, and let t ↦ φ_{t} and t ↦ ψ_{t}, for t ∈ [0, ∞), be two continuous paths of unital homomorphisms from O_{m} to D. Suppose that the unitary u_{0} = ∑_{j=1}^{m} ψ_{0}(s_{j})φ_{0}(s_{j})^{∗} satisfies [u_{0}] ∈ (m − 1)K_{1}(D). Then φ and ψ are asymptotically unitarily equivalent.

2.2.2 Lemma. (Compare with Proposition 1.7 of [35].) Let D be a unital purely infinite simple C^{∗}-algebra, with [1] = 0 in K_{0}(D). Let t ↦ φ_{t} and t ↦ ψ_{t}, for t ∈ [0, ∞), be two continuous paths of unital homomorphisms from O_{∞} to D. Then t ↦ φ_{t} and t ↦ ψ_{t} are asymptotically unitarily equivalent.

We will actually only need Lemma 2.2.1 for m = 2.

The two lemmas are actually valid for unital asymptotic morphisms (with a suitable modification of the definition of u_{0} in Lemma 2.2.2), as can be seen by applying Proposition 1.1.7. We don’t need this generality, so we don’t state it formally.

The proofs of the two lemmas are rather technical. We do the first (which is easier) in detail, and then describe the modifications needed for the second. Before proving them, we outline the basic idea. Given two continuous paths as in Lemma 2.2.1, we can start by applying Theorem 2.1.3 to the homomorphisms from O_{m} to C([0, 1]) ⊗ D obtained by restricting the paths to t ∈ [0, 1], with a certain error tolerance. The result is a continuous unitary path t ↦ v_{t}^{(1)}, for t ∈ [0, 1], such that (v_{t}^{(n)})^{∗}ψ_{t}(s_{j})v_{t}^{(n)} is close to φ_{t}(s_{j}). Extend t ↦ v_{t}^{(1)} over [0, ∞) by taking it to be constant on [1, ∞). Now replace t ↦ ψ_{t} by t ↦ γ_{t}^{(1)} = (v_{t}^{(1)})^{∗}ψ_{t}(·)v_{t}^{(1)}. Next, we want to repeat this over [1, 2], with a smaller error tolerance. Because there will be infinitely many steps, requiring in the end a product of infinitely many unitaries, there is a potential convergence problem. Thus, we insist here that the new unitary v_{t}^{(1)} be equal to 1 for t ∈ [0, 1]. In order to arrange this, we must have homomorphisms from O_{m} to C([1, 2]) ⊗ D which agree when restricted to {1} ⊂ [1, 2]. But γ_{1}^{(1)} is only close to φ_{1}. However, if γ_{1}^{(1)} is close enough to φ_{1} on the generators, then we can use exact stability of the relations to construct a new homomorphism σ : O_{m} → C([1, 2]) ⊗ D which is close to t ↦ γ_{t}^{(1)} for t ∈ [1, 2] and does agree with φ_{1} at t = 1. We apply Theorem 2.1.3 to this homomorphism, and proceed as before. Now repeat on [2, 3], etc.

In the actual proof, it is technically convenient to reduce to the case in which t ↦ φ_{t} is constant, and to start over {0} rather than over [0, 1].

The proof of Lemma 2.2.2 is essentially the same, but it is complicated by the fact that there are infinitely many generators. We deal with only finitely many of them over each interval [n, n + 1], but more and more as n increases.

Proof of Lemma 2.2.1: Corollary 2.1.13 shows that both D and C([0, 1], D) satisfy the hypotheses of Theorem 2.1.3.

By transitivity of asymptotic unitary equivalence, it suffices to show that t ↦ φ_{t} and t ↦ ψ_{t} are both asymptotically unitarily equivalent to some constant path. Thus, without loss of generality t ↦ φ_{t} is a constant path φ_{t} = φ for all t. Let λ : D → D be λ_{φ} as in Remark 2.1.2 (2).

Let f : [0, δ] → [0, ∞) be a function associated with the exact stability of O_{m} (Proposition 2.1.9) in the same way the functions f_{m} of Proposition 2.1.10 are associated with the exact stability of E_{m}.

Choose ε′_{0} > 0 with f(ε′_{0}) < 1. Choose ε_{0} > 0 with ε_{0} < 1/2, and also so small that if ω : O_{m} → A is a unital homomorphism, and a_{1}, …, a_{m} ∈ A satisfy ∥a_{j} − ω(s_{j})∥ < ε_{0}, then the a_{j} satisfy the relations for O_{m} to within ε′_{0}, that is,

∥a_{j}^{∗}a_{j} − 1∥ < ε′_{0} and ∥∑_{k=1}^{m} a_{k}a_{k}^{∗} − 1∥ < ε′_{0}

for 1 ≤ j ≤ m. Set u_{0} = ∑_{j=1}^{m} ψ_{0}(s_{j})φ(s_{j})^{∗}; this is the same as the u_{0} in the statement of the lemma, so its K_{1}-class is in (m − 1)K_{1}(D). Theorem 2.1.3 therefore yields a unitary v_{0}^{(0)} ∈ D such that ∥u_{0} − v_{0}^{(0)}λ(v_{0}^{(0)})^{∗}∥ < ε_{0}. Define v_{t}^{(0)} = v_{0}^{(0)} for all t, and define γ_{t}^{(0)} : O_{m} → D by γ_{t}^{(0)}(a) = (v_{t}^{(0)})^{∗}ψ_{t}(a)v_{t}^{(0)}. Using Remark 2.1.2, we calculate:

∥φ(s_{j}) − γ_{0}^{(0)}(s_{j})∥ = ∥v_{0}^{(0)}φ(s_{j})(v_{0}^{(0)})^{∗} − ψ_{0}(s_{j})∥
= ∥v_{0}^{(0)}λ(v_{0}^{(0)})^{∗}φ(s_{j}) − u_{0}φ(s_{j})∥ < ε_{0}

for 1 ≤ j ≤ m.

We now construct, by induction on n, numbers ε_{n}, ε′_{n} > 0 and continuous paths t ↦ v_{t}^{(n)} of unitaries in D and t ↦ γ_{t}^{(n)} of unital homomorphisms from O_{m} to D, for t ∈ [0, ∞), such that ε_{0}, ε′_{0}, v_{t}^{(0)}, and γ_{t}^{(0)} are as already chosen, and:

(1) γ_{t}^{(n)}(a) = (v_{t}^{(n)})^{∗}γ_{t}^{(n−1)}(a)v_{t}^{(n)} for a ∈ O_{m} and t ∈ [0, ∞).

(2) If n ≥ 1, then v_{t}^{(n)} = 1 for t ≤ n − 1.

(3) If n ≥ 1, then ∥φ(s_{j}) − γ_{t}^{(n)}(s_{j})∥ < 2^{−n+1} for t ∈ [n − 1, n], and if n ≥ 0 then ∥φ(s_{j}) − γ_{t}^{(n)}(s_{j})∥ < ε_{n} for t = n.

(4) f(ε′_{n}) < 2^{−n}.

(5) Whenever ω : O_{m} → A is a unital homomorphism, and a_{1}, …, a_{m} ∈ A satisfy ∥a_{j} − ω(s_{j})∥ < ε_{n}, then the a_{j} satisfy the relations for O_{m} to within ε′_{n}.

(6) ε_{n} < 2^{−(n+1)}.

Suppose that ε_{n}, ε′_{n}, v_{t}^{(n)}, and γ_{t}^{(n)} have been chosen. Choose ε′_{n+1} and then ε_{n+1} as in (4), (5), and (6).

For α ∈ [0, 1], define

a_{j}(α) = (1 − α)(φ(s_{j}) − γ_{n}^{(n)}(s_{j})) + γ_{n+α}^{(n)}(s_{j}).

Then ∥a_{j}(α)−γ_{n+α}^{(n)}(s_{j})∥ < ε_{n} for 1 ≤ j ≤ m and α ∈ [0, 1]. Conditions (4) and (5), and the choice of f, provide a unital homomorphism σ : O_{m} → C([0, 1], D) such that ∥σ(s_{j}) − a_{j}∥ < 2^{−n} for 1 ≤ j ≤ m. Define σ_{α} : O_{m} → D by σ_{α}(a) = σ(a)(α) for α ∈ [0, 1] and a ∈ O_{m}. Then

∥σ_{α}(s_{j}) − γ_{n+α}^{(n)}(s_{j})∥ < ε_{n} + 2^{−n}.

Functoriality of the approximating homomorphisms (the analog of (3) of Proposition 2.1.10) guarantees that σ_{0} = φ and σ_{1} = γ_{n+1}^{(n)}.

Define a unitary z ∈ C([0, 1], D) by z_{α} = ∑_{j=1}^{m} σ_{α}(s_{j})φ(s_{j})^{∗} for α ∈ [0, 1]. Note that z_{0} = 1, so z ∈ U_{0}(C([0, 1], D)). Theorem 2.1.3 provides a unitary α ↦ y_{α} in C([0, 1], D) such that ∥z_{α} − y_{α}λ(y_{α})^{∗}∥ < ε_{n+1}/2 for α ∈ [0, 1]. Putting α = 0, using z_{0} = 1, and rearranging terms, we obtain ∥y_{0}^{∗}λ(y_{0})−1∥ < ε_{n+1}/2. Now define

v_{t}^{(n+1)} = \{ \begin{matrix}1 & t ≤ n \\ y_{t−n}y_{0}^{∗} & n ≤ t ≤ n + 1 \\ y_{1}y_{0}^{∗} & n + 1 ≤ t \end{matrix}

and define γ_{t}^{(n+1)}(a) = (v_{t}^{(n+1)})^{∗}γ_{t}^{(n)}(a)v_{t}^{(n+1)}.

It remains only to verify condition (3) in the induction hypothesis. For α ∈ [0, 1],

∥z_{α} − v_{n+α}^{(n+1)}λ(v_{n+α}^{(n+1)})^{∗}∥ ≤ ∥z_{α} − y_{α}λ(y_{α})^{∗}∥ + ∥y_{α}∥ ∥1 − y_{0}^{∗}λ(y_{0})∥ ∥λ(y_{α})^{∗}∥
< ε_{n+1}/2 + ε_{n+1}/2 = ε_{n+1}.

Therefore, for t ∈ [n, n + 1], Remark 2.1.2 yields

∥φ(s_{j}) − γ_{t}^{(n+1)}(s_{j})∥ = ∥v_{t}^{(n+1)}φ(s_{j})(v_{t}^{(n+1)})^{∗} − γ_{t}^{(n)}(s_{j})∥
≤ ∥v_{t}^{(n+1)}λ(v_{t}^{(n+1)})^{∗}φ(s_{j}) − z_{t−n}φ(s_{j})∥ + ∥σ_{t−n}(s_{j}) − γ_{t}^{(n)}(s_{j})∥
< ε_{n+1} + ε_{n} + 2^{−n} < 2^{−(n+2)} + 2^{−(n+1)} + 2^{−n} < 2^{−n+1}.

Furthermore, if t = n + 1, then actually σ_{t−n}(s_{j}) = γ_{t}^{(n)}(s_{j}), and we obtain ∥φ(s_{j}) − γ_{t}^{(n+1)}(s_{j})∥ < ε_{n+1}. This completes the induction.

To complete the proof, we now define v_{t} = lim_{n→∞} v_{t}^{(0)}v_{t}^{(1)} ⋯ v_{t}^{(n)} for t ∈ [0, ∞). Note that the limit exists and defines a continuous unitary path t ↦ v_{t}, since v_{t}^{(n+1)}, v_{t}^{(n+2)}, … are all equal to 1 on [0, n). Furthermore, for t ∈ [n, n + 1], we have

∥φ(s_{j}) − v_{t}^{∗}ψ_{t}(s_{j})v_{t}∥ = ∥φ(s_{j}) − γ_{t}^{(n+1)}(s_{j})∥ < 2^{−n+1}.

This implies that φ and t ↦ ψ_{t} are asymptotically unitarily equivalent.

For the proof of Lemma 2.2.2, we need the following lemma.

2.2.3 Lemma. Let D be a unital purely infinite simple C^{∗}-algebra with [1] = 0 in K_{0}(D). Let m < n, and identify E_{m} with the subalgebra of O_{n} generated by s_{1}, …, s_{m}. Let φ : E_{m} → D be an injective unital homomorphism. Then there exists a unital homomorphism φ̃ : O_{n} → D such that φ̃|_{E_{m}} = φ. Moreover, if we are already given a unital homomorphism ψ : O_{n} → D, then φ̃ can be chosen to satisfy [φ̃] = [ψ] in KK^{0}(O_{n}, D).

Proof: This is essentially contained in the proof of Proposition 1.7 of [35], using the equivalence of conditions (1) and (3) in Theorem 2.1.3.

Proof of Lemma 2.2.2: We describe how to modify the proof of Lemma 2.2.1 to obtain this result.

First, note that U(D)/U_{0}(D) → K_{1}(D) is an isomorphism because D is purely infinite simple. Furthermore, cel(C([0, 1], D)) ≤ 5π/2 < ∞ by Theorem 1.2 of [62]. (It turns out that we only need this result for D = O_{∞}, so we could use Corollary 2.1.13 here instead.) Thus, the conditions on D in Lemma 2.2.1 are satisfied.

As in the proof of Lemma 2.2.1, we may assume that t ↦ φ_{t} is a constant path φ_{t} = φ for all t.

Let the functions f_{m} be the ones associated with the exact stability of E_{m} as in Proposition 2.1.10.

The proof uses an induction argument similar to that of the proof of Lemma 2.2.1, except that at the n-th stage we work with extensions to O_{2n} of φ|_{E_{n}} and ψ_{t}|_{E_{n}}. To avoid confusion, we let s_{1}, s_{2}, … be the standard generators of O_{∞}, with the first n of them generating E_{n}, and we let s_{1}^{(2n)}, …, s_{2n}^{(2n)} be the standard generators of O_{2n}, with E_{k}, for k < 2n, being identified with the subalgebra generated by the first k of them.

We start the construction at n = 2 so as not to have to worry about E_{0} and E_{1}.

In the preliminary step, we choose ε_{2} > 0 and ε′_{2} > 0 so that ε_{2} < 1/8, f_{4}(ε′_{2}) < 1/4, and whenever ω : E_{2} → A is a unital homomorphism, and a_{1}, a_{2} ∈ A satisfy ∥a_{j} − ω(s_{j})∥ < ε_{2}, then the a_{j} satisfy the relations for E_{2} to within ε′_{2}. Use Lemma 2.2.3 to choose unital homomorphisms φ^{(2)}, ψ̃_{2}^{(2)} : O_{4} → D such that φ^{(2)}|_{E_{2}} = φ|_{E_{2}}, ψ̃_{2}^{(2)}|_{E_{2}} = ψ_{2}|_{E_{2}}, and [φ^{(2)}] = [ψ̃_{2}^{(2)}] in KK^{0}(O_{4}, D). Set

u = ∑_{j=1}^{4} ψ̃_{2}^{(2)}(s_{j}^{(4)})φ^{(2)}(s_{j}^{(4)})^{∗}.

Nuclear Purely Infinite Simple C^{∗}-Algebras 73

From (2) implies (3) in Theorem 2.1.3, we obtain a unitary v_{2}^{(2)} ∈ D such that

∥u − v_{2}^{(2)}λ_{φ^{(2)}}(v_{2}^{(2)})^{∗}∥ < ε_{2}.

Define v_{t}^{(2)} = v_{2}^{(2)} for t ∈ [2, ∞), and define γ_{t}^{(2)} : O_{∞} → D by γ_{t}^{(2)}(a) = (v_{t}^{(2)})^{∗}ψ_{t}(a)v_{t}^{(2)}. As in the proof of Lemma 2.2.1, a calculation shows that

∥φ^{(2)}(s_{j}^{(4)}) − (v_{2}^{(2)})^{∗}ψ̃_{2}^{(2)}(s_{j}^{(4)})v_{2}^{(2)}∥ < ε_{2}

for 1 ≤ j ≤ 4. It follows that

∥φ(s_{j}) − γ_{2}^{(2)}(s_{j})∥ < ε_{2}

for 1 ≤ j ≤ 2.

In the induction step, we now require that t ∈ [2, ∞), that ε_{2}, ε′_{2}, γ_{t}^{(2)}, and v_{t}^{(2)} be as already given, that γ_{t}^{(n)} : O_{∞} → D, and that:

(1) γ_{t}^{(n)}(a) = (v_{t}^{(n)})^{∗}γ_{t}^{(n−1)}(a)v_{t}^{(n)} for a ∈ O_{∞} and t ∈ [2, ∞).

(2) If n ≥ 3, then v_{t}^{(n)} = 1 for t ≤ n.

(3) If n ≥ 3, then ∥φ(s_{j}) − γ_{t}^{(n)}(s_{j})∥ < 2^{−n+1} for t ∈ [n − 1, n] and 1 ≤ j ≤ n − 1, and if n ≥ 2 then ∥φ(s_{j}) − γ_{t}^{(n)}(s_{j})∥ < ε_{n} for t = n and 1 ≤ j ≤ n.

(4) f_{n}(ε′_{n}) < 2^{−n}.

(5) Whenever ω : E_{n} → A is a unital homomorphism, and a_{1}, …, a_{n} ∈ A satisfy ∥a_{j} − ω(s_{j})∥ < ε_{n}, then the a_{j} satisfy the relations for E_{n} to within ε′_{n}.

(6) ε_{n} < 2^{−(n+1)}.

For the proof of the inductive step, we first choose ε′_{n+1} and ε_{n+1} to satisfy (4), (5), and (6). Then construct, as in the proof of Lemma 2.2.1, a continuous path of homomorphisms σ_{α} : E_{n} → D such that σ_{0} = φ|_{E_{n}}, σ_{1} = γ_{n+1}^{(n)}|_{E_{n}}, and

∥σ_{α}(s_{j}) − γ_{n+α}^{(n)}(s_{j})∥ < ε_{n} + 2^{−n}

for 1 ≤ j ≤ n.

We now claim that there is a unitary path α ↦ w_{α} in D such that w_{0} = 1, w_{α}σ_{α}(s_{j}) = σ_{0}(s_{j}) for α ∈ [0, 1] and 1 ≤ j ≤ n, and w_{1}γ_{n+1}^{(n)}(s_{n+1}) = φ(s_{n+1}). To prove this, start by defining q_{α} = ∑_{j=1}^{n} σ_{α}(s_{j})σ_{α}(s_{j})^{∗}. Then set w′_{α} = ∑_{j=1}^{n} σ_{0}(s_{j})σ_{α}(s_{j})^{∗}, which is a partial isometry from q_{α} to q_{0} such that w′_{α}σ_{α}(s_{j}) = σ_{0}(s_{j}) for 1 ≤ j ≤ n. Next, define

p_{1} = γ_{n+1}^{(n)}(s_{n+1})γ_{n+1}^{(n)}(s_{n+1})^{∗} and p_{0} = φ(s_{n+1})φ(s_{n+1})^{∗}.

Documenta Mathematica 5 (2000) 49–114

Since γ_{n+1}^{(n)}|_{E_{n}} = σ_{1} and φ|_{E_{n}} = σ_{0}, we see that p_{1} and p_{0} are proper subprojections of 1 − q_{1} and 1 − q_{0} respectively, both with the same class (namely [1] = 0) in K_{0}(D). Standard methods therefore yield a unitary path α ↦ c_{α} in D such that c_{0} = 1, c_{α}q_{α}c_{α}^{∗} = q_{0}, and c_{1}p_{1}c_{1}^{∗} = p_{0}. Then φ(s_{n+1})γ_{n+1}^{(n)}(s_{n+1})^{∗}c_{1}^{∗} is a unitary in p_{0}Dp_{0}, so there is a unitary d ∈ (1 − q_{0} − p_{0})D(1 − q_{0} − p_{0}) such that

φ(s_{n+1})γ_{n+1}^{(n)}(s_{n+1})^{∗}c_{1}^{∗} + d ∈ U_{0}((1 − q_{0})D(1 − q_{0})),

and a unitary path α ↦ w′′_{α} in (1 − q_{0})D(1 − q_{0}) such that

w′′_{0} = 1 and w′′_{1} = φ(s_{n+1})γ_{n+1}^{(n)}(s_{n+1})^{∗}c_{1}^{∗} + d.

Set w_{α} = w′_{α} + w′′_{α}c_{α}; this is the path that proves the claim.

Use Lemma 2.2.3 to choose a unital homomorphism φ^{(n+1)} : O_{2n+2} → D such that φ^{(n+1)}|_{E_{n+1}} = φ|_{E_{n+1}}. Define unital homomorphisms σ̃_{α} : O_{2n+2} → D by

σ̃_{α}(s_{j}^{(2n+2)}) = w_{α}^{∗}φ^{(n+1)}(s_{j}^{(2n+2)})

for 1 ≤ j ≤ 2n + 2. Then

σ̃_{0} = φ^{(n+1)}, σ̃_{α}|_{E_{n}} = σ_{α}, and σ̃_{1}|_{E_{n+1}} = γ_{n+1}^{(n)}|_{E_{n+1}}.

Define z and choose y as in the proof of Lemma 2.2.1, using O_{2n+2} in place of O_{m}, σ̃ in place of σ, φ^{(n+1)} in place of φ, and λ = λ_{φ^{(n+1)}}. Define v_{t}^{(n+1)} and

γ_{t}^{(n+1)} as there. The same computations as there show that

∥φ(s_{j}) − γ_{t}^{(n+1)}(s_{j})∥
= ∥φ^{(n+1)}(s_{j}^{(2n+2)}) − (v_{t}^{(n+1)})^{∗}σ̃_{t−n}(s_{j}^{(2n+2)})v_{t}^{(n+1)}∥ < 2^{−n+1}

for 1 ≤ j ≤ n and t ∈ [n, n + 1], and

∥φ(s_{j}) − γ_{n+1}^{(n+1)}(s_{j})∥ = ∥φ^{(n+1)}(s_{j}^{(2n+2)}) − (v_{t}^{(n+1)})^{∗}σ̃_{1}(s_{j}^{(2n+2)})v_{t}^{(n+1)}∥ < ε_{n+1}

for 1 ≤ j ≤ n + 1. This completes the induction step.

Define v_{t} = lim_{n→∞} v_{t}^{(2)}v_{t}^{(3)} ⋯ v_{t}^{(n)}. Calculations analogous to those in the proof of Lemma 2.2.1 show that t ↦ v_{t} is a continuous unitary path in D, and that for n ≥ 2 we have

∥φ(s_{j}) − v_{t}^{∗}ψ_{t}(s_{j})v_{t}∥ < 2^{−n+1}

for t ∈ [n, n + 1] and 1 ≤ j ≤ n. This implies that

lim_{t→∞} (φ(a) − v_{t}^{∗}ψ_{t}(a)v_{t}) = 0

for all a ∈ O_{∞}.

2.2.4 Lemma. There exists a continuous family t ↦ φ_{t} of unital endomorphisms of O_{∞}, for t ∈ [0, ∞), which is asymptotically central in the sense that

lim_{t→∞} (φ_{t}(b)a − aφ_{t}(b)) = 0

for all a, b ∈ O_{∞}.

Proof: Let A_{n} be the tensor product of n copies of O_{∞}, and define μ_{n} : A_{n} → A_{n+1} by μ_{n}(a) = a ⊗ 1. Set A = lim_{⟶} A_{n}, which is just ⨂_{1}^{∞} O_{∞}. Theorem 2.1.5 implies that A_{n} ≅ O_{∞}, so Theorem 3.5 of [35] implies that A ≅ O_{∞}. (Actually, that A ≅ O_{∞} is shown in the course of the proof of Theorem 2.1.5. See [29].) It therefore suffices to construct a continuous asymptotically central inclusion of O_{∞} in A rather than in O_{∞}.

Let ν_{n} : A_{n} → A be the inclusion. Proposition 2.1.11 provides a homotopy α ↦ ψ_{α} of unital homomorphisms ψ_{α} : O_{∞} → O_{∞}⊗O_{∞} such that ψ_{0}(a) = a⊗1 and ψ_{1}(a) = 1 ⊗ a. For n ≥ 1 and t ∈ [n, n + 1], we write t = n + α and define

φ_{t}(a) = ν_{n+2}(1 ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ ψ_{α}(a)),

where the factor 1 appears n times in the tensor product. The two definitions of φ_{n}(a) agree, so t ↦ φ_{t} is continuous. We clearly have lim_{t→∞} (φ_{t}(b)a − aφ_{t}(b)) = 0 for b ∈ O_{∞} and a ∈ ⋃_{n=1}^{∞} ν_{n}(A_{n}), and a standard argument then shows this is true for all a ∈ A.

The notation introduced in the following definition is the same as in [34], [35], and [44].

2.2.5 Definition. Let A be any unital C^{∗}-algebra, and let D be a purely infinite simple C^{∗}-algebra. Let φ, ψ : A → D be two homomorphisms, and assume that φ(1) ≠ 0 and [ψ(1)] = 0 in K_{0}(D). We define a homomorphism φ˜⊕ψ : A → D, well defined up to unitary equivalence, by the following construction. Choose a projection q ∈ D such that 0 < q < φ(1) and [q] = 0 in K_{0}(D). Since D is purely infinite and simple, there are partial isometries v and w such that

vv^{∗} = φ(1) − q, v^{∗}v = φ(1), ww^{∗} = q, and w^{∗}w = ψ(1).

Now define (φ˜⊕ψ)(a) = vφ(a)v^{∗} + wψ(a)w^{∗} for a ∈ A.

The proof of the following lemma could be simplified considerably by using semiprojectivity of O_{∞} ([5]) and Proposition 1.1.7. Since [5] remains (to our knowledge) unpublished, we retain the original proof.

2.2.6 Lemma. (Compare with Proposition 2.3 of [35].) Let D be a unital purely infinite simple C^{∗}-algebra, and let q ∈ D be a projection with [q] = 0 in K_{0}(D). Let φ : O_{∞} → D and ψ : O_{∞} → qDq be unital homomorphisms. Then φ is asymptotically unitarily equivalent to φ˜⊕ψ.

Proof: Let t ↦ γ_{t} be a continuously parametrized asymptotically central inclusion of O_{∞} in O_{∞}, as in Lemma 2.2.4. Let e ∈ O_{∞} be a nonzero projection with [e] = 0 in K_{0}(O_{∞}), and set e_{t} = γ_{t}(e). Choose a continuous unitary path t ↦ u_{t} such that u_{t}e_{t}u_{t}^{∗} = e_{0}.

Let the functions f_{m} : [0, δ(m)] → [0, ∞) be as in Proposition 2.1.10. Choose numbers ε_{2} > ε_{3} > ⋯ 0 and ε′_{2} > ε′_{3} > ⋯ 0 such that:

(1) ε′_{m} < δ(m) and f_{m}(ε′_{m}) < 1/m.

(2) Whenever ω : E_{m} → A is a unital homomorphism, and a_{1}, …, a_{m} ∈ A satisfy ∥a_{j} − ω(s_{j})∥ < ε_{m}, then the a_{j} satisfy the relations for E_{m} to within ε′_{m}.

(3) ε_{m} < 1/m.

Next, use the asymptotic centrality of t ↦ e_{t} to choose t_{2} < t_{3} < ⋯, with t_{m} → ∞ as m → ∞, such that

∥s_{j} − [e_{t}s_{j}e_{t} + (1 − e_{t})s_{j}(1 − e_{t})]∥ < ε_{m}

for 1 ≤ j ≤ m and t ≥ t_{m}. Define

a_{j}(t) = u_{t} [e_{t}s_{j}e_{t} + (1 − e_{t})s_{j}(1 − e_{t})] u_{t}^{∗} ∈ e_{0}O_{∞}e_{0} ⊕ (1 − e_{0})O_{∞}(1 − e_{0}).

Conditions (1) and (2), and Proposition 2.1.10, then yield continuous paths t ↦ σ_{t}^{(m)} of homomorphisms from E_{m} to e_{0}O_{∞}e_{0} ⊕(1−e_{0})O_{∞}(1−e_{0}), defined for t ≥ t_{m}, such that ∥σ_{t}^{(m)}(s_{j})−a_{j}(t)∥ < 1/m for 1 ≤ j ≤ m, and σ_{t}^{(m+1)}|_{E_{m}} = σ_{t}^{(m)} for t ≥ t_{m+1}.

Define

α_{t}^{(m)} : E_{m} → e_{0}O_{∞}e_{0} and β_{t}^{(m)} : E_{m} → (1 − e_{0})O_{∞}(1 − e_{0})

by

α_{t}^{(m)}(a) = e_{0}σ_{t}^{(m)}(a)e_{0} and β_{t}^{(m)}(a) = (1 − e_{0})σ_{t}^{(m)}(a)(1 − e_{0}).

Note that α_{t_{m}}^{(m)} is homotopic to α_{t_{m+1}}^{(m+1)}|_{E_{m}}; since α_{t_{m+1}}^{(m+1)}|_{E_{m}} is injective, it

follows that α_{t_{m}}^{(m)} is injective. Since e_{0}O_{∞}e_{0} is purely infinite simple, it is easy to extend α_{t_{m}}^{(m)} to a homomorphism α_{t_{m}} : O_{∞} → e_{0}O_{∞}e_{0}. Proposition 2.1.11 provides homotopies t ↦ α_{t} of homomorphisms from O_{∞} to e_{0}O_{∞}e_{0}, defined for t ∈ [t_{m}, t_{m+1}], such that α_{t}|_{E_{m}} = α_{t}^{(m)} and such that α_{t_{m}} and α_{t_{m+1}} are as already given. Putting these homotopies together, and defining α_{t} = α_{t_{2}} for t ∈ [0, t_{2}], we obtain a continuous path t ↦ α_{t} of unital homomorphisms from O_{∞} to e_{0}O_{∞}e_{0}, defined for t ∈ [0, ∞), such that α_{t}|_{E_{m}} = α_{t}^{(m)} whenever t ≥ t_{m}. Similarly, there is a continuous path t ↦ β_{t} of unital homomorphisms from O_{∞} to (1 − e_{0})O_{∞}(1 − e_{0}), defined for t ∈ [0, ∞), such that β_{t}|_{E_{m}} = β_{t}^{(m)} whenever t ≥ t_{m}. Define σ_{t} : O_{∞} → O_{∞} by σ_{t}(a) = α_{t}(a) + β_{t}(a).

For t ≥ t_{m} and 1 ≤ j ≤ m, we have u_{t}^{∗}σ_{t}(s_{j})u_{t} = u_{t}^{∗}σ_{t}^{(m)}(s_{j})u_{t}, and

∥u_{t}^{∗}σ_{t}^{(m)}(s_{j})u_{t} −s_{j}∥ ≤ ∥σ_{t}^{(m)}(s_{j})−a_{j}(t)∥+∥u_{t}^{∗}a_{j}(t)u_{t} −s_{j}∥ < 1/m+ε_{m} < 2/m.

Therefore lim_{t→∞} ∥u_{t}^{∗}σ_{t}^{(m)}(s_{j})u_{t} − s_{j}∥ = 0 for all j. Thus t ↦ σ_{t} is asymptotically unitarily equivalent to id_{O_{∞}}. So φ is asymptotically unitarily equivalent to t ↦ φ ∘ σ_{t}.

Let f < φ(e_{0}) be a nonzero projection with [f] = 0 in K_{0}(D). Let w_{1}, w_{2} ∈ D be partial isometries satisfying

w_{1}^{∗}w_{1} = 1, w_{1}w_{1}^{∗} = 1 − f, and w_{1}(1 − φ(e_{0})) = (1 − φ(e_{0}))w_{1} = 1 − φ(e_{0})

and

w_{2}^{∗}w_{2} = q and w_{2}w_{2}^{∗} = f.

The homomorphism φ˜⊕ψ is only defined up to unitary equivalence, and we can take it to be

(φ˜⊕ψ)(x) = w_{1}φ(x)w_{1}^{∗} + w_{2}ψ(x)w_{2}^{∗}.

We make the same choices when defining (φ ∘ σ_{t})˜⊕ψ. Writing φ ∘ σ_{t} = φ ∘ α_{t} + φ ∘ β_{t}, with

φ ∘ α_{t} : O_{∞} → φ(e_{0})Dφ(e_{0}) and φ ∘ β_{t} : O_{∞} → φ(1 − e_{0})Dφ(1 − e_{0}),

this choice gives

(φ ∘ σ_{t})˜⊕ψ = [(φ ∘ α_{t})˜⊕ψ] + φ ∘ β_{t}.

By Lemma 2.2.2, t ↦ (φ∘α_{t})˜⊕ψ is asymptotically unitarily equivalent to φ∘α_{t}. Therefore, with ∼ denoting asymptotic unitary equivalence, we have

φ˜⊕ψ ∼ (φ ∘ α_{t})˜⊕ψ + φ ∘ β_{t} ∼ φ ∘ α_{t} + φ ∘ β_{t} ∼ φ.

This is the desired result.

2.2.7 Proposition. (Compare with Theorem 3.3 of [35].) Let D be a unital purely infinite simple C^{∗}-algebra, and let φ, ψ : O_{∞} → D be two unital homomorphisms. Then φ is asymptotically unitarily equivalent to ψ.

Proof: Let e = 1 − s_{1}s_{1}^{∗} − s_{2}s_{2}^{∗} ∈ O_{∞}, and let f = φ(e) ∈ D. Define \overline{φ} : O_{∞} → fDf by \overline{φ}(s_{j}) = φ(s_{j+2})f. Let w ∈ M_{2}(D) be a partial isometry with w^{∗}w = 1 ⊕ f and ww^{∗} = q ⊕ 0 for some q ∈ D. We regard w(φ ⊕ \overline{φ})(−)w^{∗} and w(ψ ⊕ \overline{φ})(−)w^{∗} as homomorphisms from O_{∞} to qDq. Furthermore, [q] = 0 in K_{0}(D), so

φ˜⊕w(ψ ⊕ \overline{φ})(−)w^{∗} and ψ ˜⊕w(φ ⊕ \overline{φ})(−)w^{∗}

are defined; they are easily seen to be unitarily equivalent. Using Lemma 2.2.6 for the other two steps, we therefore obtain asymptotic unitary equivalences

φ ∼ φ˜⊕w(ψ ⊕ \overline{φ})(−)w^{∗} ∼ ψ ˜⊕w(φ ⊕ \overline{φ})(−)w^{∗} ∼ ψ.

2.2.8 Corollary. Let A be any unital C^{∗}-algebra such that O_{∞} ⊗ A ≅ A. Then there exists an isomorphism β : O_{∞} ⊗ A → A such that the homomorphism a ↦ β(1 ⊗ a) is asymptotically unitarily equivalent to id_{A}.

Proof: We first prove this for A = O_{∞}. Theorem 2.1.5 implies that O_{∞} ⊗O_{∞} ≅ O_{∞}; let β_{0} : O_{∞}⊗O_{∞} → O_{∞} be an isomorphism. Then a ↦ β_{0}(1⊗a) and id_{O_{∞}} are two unital homomorphisms from O_{∞} to O_{∞}, so they are asymptotically unitarily equivalent by Proposition 2.2.7. Let t ↦ u_{t} be a unitary path such that lim_{t→∞}(β_{0}(1 ⊗ a) − u_{t}au_{t}^{∗}) = 0 for all a ∈ O_{∞}.

Now let A be as in the hypotheses. We may as well prove the result for O_{∞} ⊗ A instead of A. Take β = β_{0} ⊗ id_{A}; then a ↦ β(1 ⊗ a) is asymptotically unitarily equivalent to id_{O_{∞}⊗A} via the unitary path t ↦ u_{t} ⊗ 1.

2.3 When homotopy implies asymptotic unitary equivalence

In this subsection, we will prove that if A is a separable nuclear unital simple C^{∗}-algebra and D_{0} is unital, then two homotopic asymptotic morphisms from A to K ⊗O_{∞}⊗D_{0} are asymptotically unitarily equivalent. We will furthermore prove that an asymptotic morphism from A to K ⊗ O_{∞} ⊗ D_{0} is asymptotically unitarily equivalent to a homomorphism. The method of proof of the first statement will generalize the methods of [44]. We will obtain the second via a trick.

The following two definitions will be convenient. The first is used, both here and in Section 3, to simplify terminology, and the second is the analog of Definition 2.1 of [44].

2.3.1 Definition. Let A, D, and Q be C^{∗}-algebras, with A and Q separable and with Q also unital and nuclear. Let φ : A → D be an asymptotic morphism. A standard factorization of φ through Q ⊗ A is an asymptotic morphism ψ : Q ⊗ A → D such that φ_{t}(a) = ψ_{t}(1 ⊗ a) for all t and all a ∈ A. An asymptotic standard factorization of φ through Q ⊗ A is an asymptotic morphism ψ : Q⊗A → D such that φ is asymptotically unitarily equivalent to the asymptotic morphism (t, a) ↦ ψ_{t}(1 ⊗ a).

2.3.2 Definition. Let A, D, and φ be as in the previous definition. An (asymptotically) trivializing factorization of φ is a (asymptotic) standard factorization with Q = O_{2}. In this case, we say that φ is (asymptotically) trivially factorizable.

2.3.3 Lemma. (Compare [44], Lemma 2.2.) Let A be separable, nuclear, unital, and simple, let D_{0} be a unital C^{∗}-algebra, and let D = O_{∞} ⊗ D_{0}. Then any two full asymptotically trivially factorizable asymptotic morphisms from A to K ⊗ D are asymptotically unitarily equivalent.

Proof: It suffices to prove this for full asymptotic morphisms φ, ψ : A → K ⊗D with trivializing factorizations φ′, ψ′ : O_{2}⊗A → K ⊗D. Note that φ′ and ψ′ are again full, and it suffices to prove that φ′ is asymptotically unitarily equivalent to ψ′. By Theorem 2.1.4, we have O_{2} ⊗ A ≅ O_{2}, and Proposition 1.1.7 then implies that φ′ and ψ′ are asymptotically equal to continuous families φ′′ and ψ′′ of homomorphisms.

We now have two continuous families of full projections t ↦ φ′′_{t}(1) and t ↦ ψ′′_{t}(1) in K ⊗ D, parametrized by [0, ∞). Standard methods show that each family is unitarily equivalent to a constant projection. Moreover, the projections φ′′_{0}(1) and ψ′′_{0}(1) have trivial K_{0} classes, so are homotopic by Lemma 2.1.8 (2). Therefore they are unitarily equivalent. Combining the unitaries involved and conjugating by the result, we can assume φ′′_{t}(1) and ψ′′_{t}(1) are both equal to the constant family t ↦ p for a suitable full projection p. Now replace K ⊗ D by p(K ⊗ D)p, and apply Lemma 2.2.1; its hypotheses are satisfied by Corollary 2.1.12.

2.3.4 Corollary. (Compare [44], Lemma 2.3.) Under the hypotheses of Lemma 2.3.3, the direct sum of two full asymptotically trivially factorizable asymptotic morphisms φ, ψ : A → K ⊗ D is again full and asymptotically trivially factorizable.

Proof: Since asymptotic unitary equivalence respects direct sums, the previous lemma implies we may assume φ = ψ. We may further assume that φ actually has a trivializing factorization φ′ : O_{2} ⊗ A → K ⊗ D. Then φ ⊕ ψ has the standard factorization id_{M_{2}} ⊗φ′ through (M_{2}⊗O_{2})⊗A, and this is a trivializing factorization because M_{2} ⊗ O_{2} ≅ O_{2}.

Fullness follows from Lemma 1.2.6 (2).

We also need asymptotically standard factorizations through O_{∞} ⊗A. The special properties required in the following lemma will be used in the proof of Theorem 2.3.7.

2.3.5 Lemma. Let A be a separable unital nuclear C^{∗}-algebra, let D_{0} be unital, and let D = O_{∞} ⊗D_{0}. Let φ : A → K ⊗D be an asymptotic morphism. Then φ has an asymptotic standard factorization through O_{∞} ⊗ A. In fact, φ is asymptotically unitarily equivalent to an asymptotic morphism of the form ψ_{t}(a) = δ ∘ (id_{O_{∞}} ⊗ φ̃_{t})(1 ⊗ a), in which δ : O_{∞} ⊗ K ⊗ D → K ⊗ D is an isomorphism, φ̃ is completely positive contractive and asymptotically equal to φ, and id_{O_{∞}} ⊗ φ̃_{t} is defined to be the tensor product of completely positive maps and is again completely positive contractive.

Proof: Lemma 1.1.5 provides a completely positive contractive asymptotic morphism φ̃ which is asymptotically equal to φ. Then id_{O_{∞}} ⊗φ̃_{t} is the minimal tensor product of two completely positive contractive linear maps, and is therefore bounded and completely positive by Proposition IV.4.23 (i) of [58]. Looking at the proof of that proposition and of Theorem IV.3.6 of [58], we see that such a tensor product is in fact contractive. Thus, ∥id_{O_{∞}} ⊗ φ̃_{t}∥ ≤ 1 for all t. One checks that t ↦ (id_{O_{∞}} ⊗ φ̃_{t})(b) is continuous for b in the algebraic tensor product of O_{∞} and A. It follows that continuity holds for all b ∈ O_{∞} ⊗ A. Similarly, one checks that t ↦ id_{O_{∞}} ⊗ φ̃_{t} is asymptotically multiplicative, so is an asymptotic morphism.

Use Corollary 2.2.8 to find an isomorphism δ_{0} : O_{∞} ⊗ D → D such that d ↦ δ_{0}(1 ⊗ d) is asymptotically unitarily equivalent to id_{D}. This induces an isomorphism δ : O_{∞} ⊗ K ⊗ D → K ⊗ D, and a unitary path t ↦ u_{t}^{(0)} ∈ M(K ⊗D) such that ∥u_{t}^{(0)}δ(1⊗d)(u_{t}^{(0)})^{∗}−d∥ → 0 for all d ∈ K ⊗D. By Lemma 1.3.9, there is a unitary path t ↦ u_{t} ∈ (K⊗D)^{+} such that ∥u_{t}δ(1⊗d)u_{t}^{∗}−d∥ → 0 for all d ∈ K ⊗ D.

We prove that the ψ that results from these choices is in fact asymptotically unitarily equivalent to φ̃; this will prove the lemma. Choose finite subsets F_{1} ⊂ F_{2} ⊂ ⋯ whose union is dense in A. For each n, note that the set S_{n} = {φ̃_{t}(a) : a ∈ F_{n}, t ∈ [0, n]} is compact in D, so that there is r_{n} with ∥u_{t}δ(1 ⊗ d)u_{t}^{∗} − d∥ < 2^{−n} for all d ∈ S_{n} and t ≥ r_{n}. For α ∈ [0, 1] define f(n+α) = (1−α)r_{n}+αr_{n+1}. Then define unitaries v_{t} ∈ (K⊗D)^{+} by v_{t} = u_{f(t)}. For t ∈ [n, n + 1] and a ∈ F_{n}, this gives (using f(t) ≥ r_{n})

∥v_{t}ψ_{t}(a)v_{t}^{∗} − φ̃_{t}(a)∥ = ∥u_{f(t)}δ(1 ⊗ φ̃_{t}(a))u_{f(t)}^{∗} − φ̃_{t}(a)∥ < 2^{−n}.

Thus ψ is in fact asymptotically unitarily equivalent to φ̃.

2.3.6 Lemma. (Compare [44], Proposition 3.3.) Assume the hypotheses of Lemma 2.3.3. Let φ, ψ : A → K ⊗ D be full asymptotic morphisms with ψ asymptotically trivially factorizable. Then φ ⊕ ψ is asymptotically unitarily equivalent to φ.

Proof: By Lemma 2.3.5, we may assume that φ has a standard factorization through O_{∞} ⊗ A, say φ′ : O_{∞} ⊗ A → K ⊗ D. Using Lemma 1.3.8 on φ′ and on an asymptotically trivializing factorization for ψ, we may assume without loss of generality that there are projections p, q ∈ K ⊗ D such that φ′ is a unital asymptotic morphism from A to p(K ⊗D)p and ψ is an asymptotically trivially factorizable unital asymptotic morphism from A to q(K ⊗ D)q.

Choose a nonzero projection e ∈ O_{∞} with trivial K_{0} class. Let t ↦ f_{t} be a tail projection for φ′(e ⊗ 1). Choose a continuous unitary family t ↦ u_{t} in p(K ⊗ D)p such that u_{t}f_{t}u_{t}^{∗} = f_{0} for all t. Define bounded asymptotic morphisms

σ : (1 − e)O_{∞}(1 − e) ⊗ A → (p − f_{0})(K ⊗ D)(p − f_{0})

and

τ : eO_{∞}e ⊗ A → f_{0}(K ⊗ D)f_{0}

by

σ_{t}(x) = u_{t}(p − f_{t})φ′_{t}(x)(p − f_{t})u_{t}^{∗} and τ_{t}(x) = u_{t}f_{t}φ′_{t}(x)f_{t}u_{t}^{∗}.

These are in fact asymptotic morphisms, because lim_{t→∞} ∥f_{t} − φ′_{t}(e ⊗ 1)∥ = 0. Then define asymptotic morphisms

σ̃ : A → (p − f_{0})(K ⊗ D)(p − f_{0}) and τ̃: A → f_{0}(K ⊗ D)f_{0}

by

σ̃_{t}(a) = σ_{t}((1 − e) ⊗ a)) and τ̃_{t}(a) = τ_{t}(e ⊗ a).

It follows that

lim_{t→∞} ∥u_{t}φ′_{t}(1 ⊗ a)u_{t}^{∗} − σ̃_{t}(a) − τ̃_{t}(a)∥ = 0

for all a ∈ A, so φ is asymptotically unitarily equivalent to σ̃ ⊕ τ̃. Since [e] = 0 in K_{0}(O_{∞}), there is a unital homomorphism ν : O_{2} → eO_{∞}e, and the formula τ̃_{t}(a) = (τ_{t} ∘ (ν ⊗ id_{A}))(1 ⊗ a) shows that τ̃ has a trivializing factorization. Furthermore, τ̃ is full because φ′ is. So τ̃ ⊕ ψ is full and asymptotically trivially factorizable by Corollary 2.3.4, and therefore asymptotically unitarily equivalent to τ̃ by Lemma 2.3.3. The asymptotic unitary equivalence of φ and σ̃ ⊕ τ̃ now implies that φ ⊕ ψ is asymptotically unitarily equivalent to φ.

We now come to the main technical theorem of this section.

2.3.7 Theorem. Let A be separable, nuclear, unital, and simple. Let D_{0} be a unital C^{∗}-algebra, and let D = O_{∞} ⊗ D_{0}. Then two full asymptotic morphisms from A to K ⊗ D are asymptotically unitarily equivalent if and only if they are homotopic.

This result is a continuous analog of Theorem 3.4 of [44], which gives a similar result for approximate unitary equivalence. In the proof of that theorem, to get approximate unitary equivalence to within ε on a finite set F, it was necessary to approximately absorb a large direct sum of asymptotically trivially factorizable homomorphisms—a direct sum which had to be larger for smaller ε and larger F. In the proof of the theorem stated here, we must continuously interpolate between approximate absorption of ever larger numbers of asymptotic morphisms. The resulting argument is rather messy. We try to make it easier to follow by isolating two pieces as lemmas. For the first of these, recall from Definition 1.1.4 (2) that bounded asymptotic morphisms are assumed in particular to be linear.

2.3.8 Lemma. Let A and D be C^{∗}-algebras, with A separable. Let α ↦ φ^{(α)} be a bounded homotopy of asymptotic morphisms from A to D. Then there exists a continuous function f : [0, ∞) → (0, ∞) such that for every a ∈ A, we have

lim_{t→∞} ( sup_{|α_{1}−α_{2}|≤1/f(t)} ∥φ_{t}^{(α_{1})}(a) − φ_{t}^{(α_{2})}(a)∥) = 0.

Proof: Choose finite sets F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ A whose union is dense in A.

For each n and each fixed a ∈ A, the map (t, α) ↦ φ_{t}^{(α)}(a) is uniformly continuous on [0, n] × [0, 1]. So there is δ_{n} > 0 such that

sup{∥φ_{t}^{(α_{1})}(a) − φ_{t}^{(α_{2})}(a)∥ : t ∈ [0, n], |α_{1} − α_{2}| ≤ δ_{n}, a ∈ F_{n}} < 2^{−n}.

We may clearly assume δ_{1} ≥ δ_{2} ≥ · · · . Let t ↦ δ(t) be a continuous function such that 0 < δ(t) ≤ δ_{n} for t ∈ [n − 1, n].

We claim that if a ∈ ⋃_{n=0}^{∞} F_{n}, then

lim_{t→∞} ( sup_{|α_{1}−α_{2}|≤δ(t)} ∥φ_{t}^{(α_{1})}(a) − φ_{t}^{(α_{2})}(a)∥) = 0.

To see this, let a ∈ F_{m}. For n ≥ m + 1, t ∈ [n − 1, n], and |α_{1} − α_{2}| ≤ δ(t), we have in particular |α_{1} − α_{2}| ≤ δ_{n}, so that ∥φ_{t}^{(α_{1})}(a) − φ_{t}^{(α_{2})}(a)∥ ≤ 2^{−n}.

The statement of the lemma, using f(t) = 1/δ(t), follows from the claim by a standard argument, since φ is bounded and ⋃_{n=0}^{∞} F_{n} is dense in A.

2.3.9 Lemma. Let A and Q be C^{∗}-algebras, with Q unital and nuclear. Let N ≥ 2, let e_{0}, e_{1}, …, e_{N} ∈ Q be mutually orthogonal projections which sum to 1, and let w ∈ Q be a unitary such that we_{0}w^{∗} ≤ e_{1}, we_{j}w^{∗} ≤ e_{j} + e_{j+1} for 1 ≤ j ≤ N − 1, and we_{N}w^{∗} ≤ e_{N} + e_{0}. Let a_{0}, …, a_{N}, b_{0}, …, b_{N} ∈ A. Then in Q ⊗ A we have

∥(w ⊗ 1) (∑_{j=0}^{N} e_{j} ⊗ a_{j}) (w ⊗ 1)^{∗} − ∑_{j=0}^{N} e_{j} ⊗ b_{j}∥

≤ max{∥a_{N} − b_{0}∥, ∥a_{0} − b_{1}∥ + ∥a_{1} − b_{1}∥, ∥a_{1} − b_{2}∥ + ∥a_{2} − b_{2}∥,
…, ∥a_{N−1} − b_{N}∥ + ∥a_{N} − b_{N}∥}.

Proof: Let

x = (w ⊗ 1) (∑_{j=0}^{N} e_{j} ⊗ a_{j}) (w ⊗ 1)^{∗} = ∑_{j=0}^{N} we_{j}w^{∗} ⊗ a_{j} and y = ∑_{j=0}^{N} e_{j} ⊗ b_{j}.

Observe that if we take the indices mod N + 1, then e_{k} is orthogonal to we_{j}w^{∗} whenever k ≠ j, j + 1, and also if j = k = 0. Therefore we can calculate

x − y = (∑_{i=0}^{N} e_{i} ⊗ 1) (x − y) (∑_{k=0}^{N} e_{k} ⊗ 1)

= ∑_{j=0}^{N} [(e_{j} ⊗ 1)(we_{j}w^{∗} ⊗ a_{j} + we_{j−1}w^{∗} ⊗ a_{j−1})(e_{j} ⊗ 1) − e_{j} ⊗ b_{j}]

+ ∑_{j=0}^{N} [e_{j}(we_{j}w^{∗})e_{j+1} + e_{j+1}(we_{j}w^{∗})e_{j}] ⊗ b_{j}.

We now claim that the second term in the last expression is zero. The projections we_{k}w^{∗} are orthogonal and add up to 1, and e_{j+1} is orthogonal to all of them except for k = j and k = j +1. Therefore e_{j+1} ≤ we_{j}w^{∗} +we_{j+1}w^{∗}. Also, e_{j}we_{j+1}w^{∗} = 0, so we obtain

e_{j}(we_{j}w^{∗})e_{j+1} = e_{j}(we_{j}w^{∗} + we_{j+1}w^{∗})e_{j+1} = e_{j}e_{j+1} = 0.

Similarly, e_{j+1}(we_{j}w^{∗})e_{j} = 0. So the claim is proved.

It remains to estimate the first term. Since the summands are orthogonal, the norm of this term is bounded by the maximum of the norms of the summands. Using again e_{j} ≤ we_{j−1}w^{∗} + we_{j}w^{∗}, we obtain

∥(e_{j} ⊗ 1)(we_{j}w^{∗} ⊗ a_{j} + we_{j−1}w^{∗} ⊗ a_{j−1})(e_{j} ⊗ 1) − e_{j} ⊗ b_{j}∥

≤ ∥a_{j−1} − a_{j}∥ + ∥(e_{j} ⊗ 1)(we_{j}w^{∗} ⊗ a_{j} + we_{j−1}w^{∗} ⊗ a_{j})(e_{j} ⊗ 1) − e_{j} ⊗ b_{j}∥
= ∥a_{j−1} − a_{j}∥ + ∥e_{j} ⊗ (a_{j} − b_{j})∥ ≤ ∥a_{j−1} − a_{j}∥ + ∥a_{j} − b_{j}∥.

If j = 0, then j − 1 = N. We then have also e_{0}we_{0}w^{∗} = 0, so e_{0} ≤ we_{N}w^{∗}, whence

∥(e_{0} ⊗ 1)(we_{0}w^{∗} ⊗ a_{0} + we_{N}w^{∗} ⊗ a_{N})(e_{0} ⊗ 1) − e_{0} ⊗ b_{0}∥
= ∥e_{0} ⊗ (a_{N} − b_{0})∥ ≤ ∥a_{N} − b_{0}∥.

This proves the lemma.

Proof of Theorem 2.3.7: That asymptotic unitary equivalence implies homotopy is Lemma 1.3.3 (1). We therefore prove the reverse implication.

Using Lemma 2.3.5, we may without loss of generality assume our homotopy has the form φ_{t}^{(α)}(a) = δ(1_{O_{∞}} ⊗φ_{t}^{(α)}(a)), where δ : O_{∞}⊗K ⊗ D → K ⊗D is a homomorphism and φ is a completely positive contractive asymptotic morphism from A to C([0, 1], K ⊗ D). It then suffices to prove the theorem for the homotopy of asymptotic morphisms from A to O_{∞} ⊗ K ⊗ D given by \overline{φ}_{t}^{(α)}(a) = 1 ⊗ φ_{t}^{(α)}(a). (We get an asymptotic unitary equivalence of φ^{(0)} and φ^{(1)} by applying δ.)

The next step is to do some constructions in O_{∞} and O_{2}. Choose a projection e ∈ O_{∞} with e ≠ 1 and [e] = [1] in K_{0}(O_{∞}). Choose a unital homomorphism γ : O_{2} → (1−e)O_{∞}(1−e). Define isometries s̃_{j} ∈ O_{∞} by s̃_{j} = γ(s_{j}). Let λ : O_{2} → O_{2} be the standard shift λ(c) = s_{1}cs_{1}^{∗}+s_{2}cs_{2}^{∗}. Since any two unital endomorphisms of O_{2} are homotopic (by Remark 2.1.2 (1) and the connectedness of the unitary group of O_{2}), there is a homotopy α ↦ ω_{α} of endomorphisms of O_{2} with ω_{0} = id_{O_{2}} and ω_{1} = λ.

We will now suppose that we are given continuous functions

α_{n} : [n − 1, ∞) → [0, 1]

for n ≥ 1 such that

α_{n+1}(n) = α_{n}(n) (1)

for all n, and a continuous function F : [0, ∞) → (0, ∞). (These will be chosen below.) Then we define ψ_{t} : O_{2} ⊗ A → O_{∞} ⊗ K ⊗ D by

ψ_{t}(c ⊗ a) = ∑_{k=1}^{n} s_{2}^{k−1}s̃_{1}γ(c)(s_{2}^{k−1}s̃_{1})^{∗} ⊗ φ_{t}^{(α_{k}∘F (t))}(a)

+ s_{2}^{n}γ(ω_{F (t)−n}(c))(s_{2}^{n})^{∗} ⊗ φ_{t}^{(α_{n+1}∘F (t))}(a) for F (t) ∈ [n, n + 1]. (2)

(This is an orthogonal sum since the projections

s̃_{1}s_{1}^{∗}, s̃_{2}s̃_{1}(s̃_{2}s̃_{1})^{∗}, …, s_{2}^{n−1}s̃_{1}(s_{2}^{n−1}s̃_{1})^{∗}, s_{2}^{n}(s_{2}^{n})^{∗}

are mutually orthogonal.) As in the proof of Lemma 2.3.5, each ψ_{t} is well defined, linear, and contractive, and t ↦ ψ_{t}(b) is continuous for b in the algebraic tensor product of O_{2} and A (using (1) when F (t) ∈ N), and so for all b ∈ O_{2} ⊗ A.

We now claim that ψ, as defined by (2), is actually an asymptotic morphism from O_{2} ⊗ A to O_{∞} ⊗ K ⊗ D. It only remains to prove asymptotic multiplicativity. By linearity and finiteness of sup_{t} ∥ψ_{t}∥, it suffices to do this on elementary tensors. Since γ, ω_{α}, and the maps c ↦ s_{2}^{k−1}s̃_{1}c(s_{2}^{k−1}s̃_{1})^{∗} and c ↦ s_{2}^{n}c(s_{2}^{n})^{∗} are homomorphisms (and so contractive), a calculation gives, for F (t) ∈ [n, n + 1],

lim_{t→∞} ∥ψ_{t}((c_{1} ⊗ a_{1})(c_{2} ⊗ a_{2})) − ψ_{t}(c_{1} ⊗ a_{1})ψ_{t}(c_{2} ⊗ a_{2})∥
≤ lim_{t→∞} ∥c_{1}c_{2}∥ ( sup_{α∈[0,1]} ∥φ_{t}^{(α)}(a_{1}a_{2}) − φ_{t}^{(α)}(a_{1})φ_{t}^{(α)}(a_{2})∥) = 0.

Define ι : A → O_{2}⊗A by ι(a) = 1⊗a. Then ψ∘ι is an asymptotic morphism from A to O_{∞} ⊗ K ⊗ D. By definition, it has a trivializing factorization, so Lemma 2.3.6 implies that \overline{φ}^{(α)} ⊕ (ψ ∘ ι) is asymptotically unitarily equivalent to \overline{φ}^{(α)}. The theorem will therefore be proved if we can choose the functions F and α_{n} in such a way that \overline{φ}^{(0)} ⊕ (ψ ∘ ι) is asymptotically unitarily equivalent to \overline{φ}^{(1)} ⊕ (ψ ∘ ι).

Before actually choosing F and the α_{n}, we construct, in terms of F, the unitary path we will use for the desired asymptotic unitary equivalence. Let τ be an automorphism of M_{2}(O_{∞}) which sends 1 ⊕ 0 to e ⊕ 0 and 0 ⊕ c to c ⊕ 0 for all c ∈ (1 − e)O_{∞}(1 − e). Let τ̃ be the obvious induced automorphism of M_{2}(O_{∞} ⊗ K ⊗ D). It suffices to prove asymptotic unitary equivalence of τ̃ ∘ (\overline{φ}^{(0)} ⊕ (ψ ∘ ι)) and τ̃ ∘ (\overline{φ}^{(1)} ⊕ (ψ ∘ ι)). Furthermore, these two asymptotic morphisms take values in O_{∞} ⊗ K ⊗ D, embedded as the upper left corner, so we only work there. This results in the identification

τ̃ ∘ (\overline{φ}^{(α)} ⊕ (ψ ∘ ι)) = e ⊗ φ^{(α)}(−) + (ψ ∘ ι).

We further note that, by Lemma 1.3.9, it suffices to construct a continuous family of unitaries in the multiplier algebra M(O_{∞} ⊗ K ⊗ D).

With these identifications and reductions, our unitary path will take the form u_{t} = v(F (t)) ⊗ 1 for a suitable unitary path r ↦ v(r) in O_{∞}, defined for r ∈ [0, ∞). The construction of v requires further notation.

Define projections in O_{∞} by p_{k} = s_{2}^{k−1}s̃_{1}(s_{2}^{k−1}s̃_{1})^{∗} and q_{n} = s_{2}^{n}(s_{2}^{n})^{∗}. Then

p_{1} + p_{2} + ⋯ + p_{n} + q_{n} + e = 1 and p_{n+1} + q_{n+1} = q_{n}.

Choose projections f_{k} < p_{k} with [f_{k}] = 1 in K_{0}(O_{∞}). Note that f_{n+1} < q_{n}. Then there are partial isometries v_{k} with

v_{0}^{∗}v_{0} = e, v_{0}v_{0}^{∗} = f_{1}, v_{k}^{∗}v_{k} = f_{k}, and v_{k}v_{k}^{∗} = f_{k+1},

and w_{n} with

w_{n}^{∗}w_{n} = f_{n+1} and w_{n}w_{n}^{∗} = e.

Using the connectedness of the unitary group of (f_{n+1}+f_{n+2})O_{∞}(f_{n+1}+f_{n+2}), choose a continuous path α ↦ y_{n}(α) of partial isometries from f_{n+1} + f_{n+2} to f_{n+2} + e such that y_{n}(0) = w_{n} + f_{n+2} and y_{n}(1) = v_{n+1} + w_{n+1}. Then define

v(n+α) = (p_{1}−f_{1})+⋯+(p_{n+1}−f_{n+1})+(q_{n+1}−f_{n+2})+v_{0}+v_{1}+⋯+v_{n}+y_{n}(α)

for n ∈ N and α ∈ [0, 1]. There are two definitions at each integer, but they agree, so v is a continuous path of unitaries. Furthermore, one immediately verifies that for fixed r ∈ [n, n + 1], the unitary w = v(r) and sequence of projections

e_{0} = e, e_{1} = p_{1}, e_{2} = p_{2}, …, e_{n} = p_{n}, e_{n+1} = q_{n} (3)

satisfy the hypotheses in Lemma 2.3.9.

Now take f to be as in Lemma 2.3.8, and set F (t) = f(t) + 2. Define α_{0} : [0, ∞) → [0, 1] by α_{0}(r) = 0 for all r, and choose the functions α_{n} : [n − 1, ∞) → [0, 1] to be continuous, to satisfy (1), and such that α_{n+1}(r) = 1 for r ∈ [n, n + 1] and

|α_{k+1}(r) − α_{k}(r)| ≤ 1/(n − 1) for r ∈ [n, n + 1] and 0 ≤ k ≤ n.

Take ψ and u to be defined using these choices of F and the α_{n}. Let t ∈ [0, ∞). Set r = F (t) and choose n ∈ N such that r ∈ [n, n + 1]. Let w = v(r) and let e_{0}, …, e_{n+1} be as in (3). For a ∈ A, we then have

∥u_{t} [e ⊗ φ^{(0)}(a) + ψ_{t}(1 ⊗ a)] u_{t}^{∗} − [e ⊗ φ^{(1)}(a) + ψ_{t}(1 ⊗ a)]∥
= ∥(w ⊗ 1) [∑_{k=0}^{n+1} e_{k} ⊗ φ_{t}^{(α_{k}(r))}(a)] (w ⊗ 1)^{∗}

− [e_{0} ⊗ φ_{t}^{(1)}(a) + ∑_{k=1}^{n+1} e_{k} ⊗ φ_{t}^{(α_{k}(r))}(a)]∥ .

Apply Lemma 2.3.9 with a_{k} = b_{k} = φ_{t}^{(α_{k}(r))}(a) for 1 ≤ k ≤ n + 1, and with a_{0} = φ_{t}^{(α_{0}(r))}(a) and b_{0} = φ_{t}^{(1)}(a) = φ_{t}^{(α_{n+1}(r))}(a) = a_{n+1}. It follows that the expression above is at most

max(0, ∥a_{0} − a_{1}∥, …, ∥a_{n} − a_{n+1}∥)
= max{∥φ_{t}^{(α_{k}(r))}(a) − φ_{t}^{(α_{k+1}(r))}(a)∥ : 0 ≤ k ≤ n}
≤ sup{∥φ_{t}^{(α_{1})} − φ_{t}^{(α_{2})}∥ : |α_{1} − α_{2}| ≤ 1/(n − 1)}.

Since n − 1 ≥ r − 2 = f(t), we have 1/(n − 1) ≤ 1/f(t), and this last expression converges to 0 as t → ∞. Thus we have shown that

e ⊗ φ^{(0)}(−) + (ψ ∘ ι) and e ⊗ φ^{(1)}(−) + (ψ ∘ ι)

are asymptotically unitarily equivalent. This completes the proof.

2.3.10 Corollary. Let A be separable, nuclear, unital, and simple, let D_{0} be unital, and let D = O_{∞} ⊗ D_{0}. Then any full asymptotic morphism φ : A → K ⊗ D is asymptotically unitarily equivalent to a homomorphism.

Proof: It is obvious that an asymptotic morphism is homotopic to all of its reparametrizations. The result therefore follows from Theorem 2.3.7 and Proposition 1.3.7.

2.3.11 Remark. The hypothesis of fullness can be removed in Theorem 2.3.7 (and in Corollary 2.3.10) in the following way. Let α ↦ φ^{(α)} be a homotopy of asymptotic morphisms from A to K ⊗D, with D = O_{∞} ⊗D_{0}. Applying Lemma 1.3.8, we can assume α ↦ φ^{(α)} is a homotopy of unital (hence full) asymptotic morphisms from A to D′ = p(K ⊗ D)p for a suitable projection p. The algebra D′ is stable under tensoring with O_{∞} by Corollary 2.1.12. So we can apply the result already proved to asymptotic morphisms from A to K ⊗ D′. Then embed K ⊗ D′ in K ⊗ D.

3 Unsuspended E-theory for simple nuclear C^{∗}-algebras

In [16], Dǎdǎrlat and Loring proved that for certain C^{∗}-algebras A, one can obtain the groups KK^{0}(A, B) via “unsuspended E-theory”: KK^{0}(A, B) ≅ [[K ⊗ A, K ⊗ B]] (notation from Definition 1.1.2) for all separable B. The terminology comes from the omission of the suspension that is normally required. The conditions on A are quite restrictive, and in particular fail for trivial reasons as soon as A has even one nonzero projection.

In this section, we want to take A to be separable, nuclear, unital, and simple. To make enough room, we assume B is a tensor product O_{∞} ⊗ D with D unital. We then discard the class of the zero asymptotic morphism (the source of the difficulty with projections). We are able to prove, with the help of Kirchberg’s results as stated in Section 2.1 and also using Theorem 2.3.7, that we do in fact get KK^{0}(A, B) as a set of suitable homotopy classes of asymptotic morphisms from K ⊗A to K ⊗B. (Corollary 2.3.10 implies that we can even use asymptotic unitary equivalence classes of homomorphisms. See Section 4.1.)

In the first subsection, we construct for fixed A a middle exact homotopy invariant functor from separable C^{∗}-algebras to abelian groups in a manner analogous to the definition of K_{0}(D), but using asymptotic morphisms from A to K ⊗ O_{∞} ⊗ D^{+} in place of projections in K ⊗ D^{+}. The fact that the target algebra is infinite means that, as for K_{0} of a purely infinite simple C^{∗}-algebra, we do not need to take formal differences of classes. We do, however, need to introduce the unitization of the target algebra for essentially the same reason that it is necessary in the definition of K_{0}. In the second subsection, we then show that this functor is naturally isomorphic to KK^{0}(A, −).

3.1 The groups [[A, K ⊗ O_{∞} ⊗ D]]_{+} and Ẽ_{A}(D)

Let A be separable, nuclear, unital, and simple. In this subsection we construct a functor [[A, K ⊗ O_{∞} ⊗ −]]_{+} on unital C^{∗}-algebras and the corresponding functor Ẽ_{A}(−) on general C^{∗}-algebras (obtained via the unitization). We then prove that Ẽ_{A} is a cohomology theory on separable C^{∗}-algebras in the usual sense. This information is needed in order to apply the uniqueness theorems for KK-theory in the next subsection.

3.1.1 Definition. Let A be separable and unital, and assume each ideal of A is generated by its projections. Let B have an approximate identity of projections. Then [[A, B]]_{+} denotes the set of homotopy classes of full asymptotic morphisms from A to B.

3.1.2 Proposition. Let A be simple, separable, unital, and nuclear. For any unital C^{∗}-algebra D, give [[A, K ⊗ O_{∞} ⊗ D]]_{+} the addition operation that it receives from being a subset of [[A, K ⊗ O_{∞} ⊗ D]]. Then [[A, K ⊗ O_{∞} ⊗ −]]_{+} is a functor from separable unital C^{∗}-algebras and homotopy classes of unital asymptotic morphisms to abelian groups. The zero element is the class of any full asymptotic morphism from A to K ⊗O_{∞} ⊗D with a standard factorization (see Definition 2.3.1) through O_{2} ⊗ A.

Proof: Lemma 1.2.6 (2) shows that [[A, K ⊗ O_{∞} ⊗ D]]_{+} is closed under the addition in [[A, K ⊗ O_{∞} ⊗ D]]. Therefore [[A, K ⊗ O_{∞} ⊗ D]]_{+} is an abelian semigroup, provided it is not empty.

According to Theorem 2.3.7, homotopy is the same relation as asymptotic unitary equivalence in this set. So we can use them interchangeably.

For functoriality, let E be another unital C^{∗}-algebra, and let φ : D → E be a unital asymptotic morphism. Let \overline{φ} = id_{K⊗O_{∞}} ⊗ φ (see Proposition 1.1.8) be the induced asymptotic morphism from K ⊗ O_{∞} ⊗ D to K ⊗ O_{∞} ⊗ E. It is full because if e ∈ K is any nonzero projection, then e ⊗ 1 ⊗ 1 is a full projection in K ⊗ O_{∞} ⊗ D which is sent to the full projection e ⊗ 1 ⊗ 1 in K ⊗ O_{∞} ⊗ E. Lemmas 1.2.6 (2) and 2.1.8 (1) now imply that η ↦ [[φ]] · η sends full asymptotic morphisms to full asymptotic morphisms.

We now construct an identity element. Theorem 2.1.4 provides an isomorphism ν : O_{2} ⊗ A → O_{2}. Let τ : O_{2} → O_{∞} be an injective homomorphism (sending 1 to a nonzero projection in O_{∞} with trivial K_{0}-class), and define a full homomorphism ζ : A → O_{∞} by ζ(a) = (τ ∘ ν)(1 ⊗ a). Composing it with the full homomorphism x ↦ e ⊗ x ⊗ 1 from O_{∞} to K ⊗ O_{∞} ⊗ D, where e ∈ K is any nonzero projection, we obtain a full asymptotic morphism from A to K ⊗ O_{∞} ⊗ D which has a standard factorization through O_{2} ⊗ A.

Lemma 2.3.3 implies that any other full asymptotic morphism with a trivializing factorization is asymptotically unitarily equivalent to ζ. This class acts as the identity by Lemma 2.3.6.

Finally, we must construct additive inverses. Let η ∈ [[A, K ⊗ O_{∞} ⊗ D]]_{+}. By Lemma 2.3.5, we can take η = [[φ]], where φ has a standard factorization through O_{∞} ⊗ A, say φ_{t}(a) = ψ_{t}(1 ⊗ a) for some asymptotic morphism ψ : O_{∞}⊗A → K⊗O_{∞}⊗D. Choose a projection f ∈ O_{∞} with [f] = −1 in K_{0}(O_{∞}). Define \overline{ψ}_{t} = ψ_{t}|_{fO_{∞}f⊗A}, and define \overline{φ} : A → K ⊗O_{∞} ⊗D by \overline{φ}_{t}(a) = φ_{t}(f ⊗a). Choose a unital homomorphism

ν : O_{2} → ( \begin{matrix}1 & 0 \\ 0 & f \end{matrix} ) M_{2}(O_{∞}) ( \begin{matrix}1 & 0 \\ 0 & f \end{matrix} ) .

Then (id_{M_{2}} ⊗ ψ) ∘ ν provides a standard factorization of φ ⊕ \overline{φ} through O_{2} ⊗ A. Note that φ ⊕ \overline{φ} is full because φ is, so it is asymptotically unitarily equivalent to ζ by Lemma 2.3.3. This shows that [[\overline{φ}]] is the inverse of η.

3.1.3 Definition. If D is any C^{∗}-algebra, then we denote by D^{#} the C^{∗}- algebra K ⊗ O_{∞} ⊗ D^{+}. We use the analogous notation for homomorphisms. If D is separable, we define Ẽ_{A}(D) to be the kernel of the map [[A, D^{#}]]_{+} → [[A, K ⊗ O_{∞}]]_{+} induced by the unitization map D^{+} → C.

3.1.4 Proposition. Let A be separable, nuclear, unital, and simple. Then Ẽ_{A} is a functor from separable C^{∗}-algebras and homotopy classes of asymptotic morphisms to abelian groups.

Proof: This follows from Proposition 3.1.2 and the fact that unitizations and tensor products of asymptotic morphisms are well defined (Lemma 1.1.6 and Proposition 1.1.8).

3.1.5 Remark. It is obvious that if D_{1} and D_{2} are unital, then there is a natural isomorphism

[[A, K ⊗ O_{∞} ⊗ (D_{1} ⊕ D_{2})]]_{+} ≅ [[A, K ⊗ O_{∞} ⊗ D_{1}]]_{+} ⊕ [[A, K ⊗ O_{∞} ⊗ D_{2})]]_{+}.

It follows that for unital D, there is a natural isomorphism

Ẽ_{A}(D) ≅ [[A, K ⊗ O_{∞} ⊗ D]]_{+}.

We will sometimes denote by φ_{∗} the map [[A, D_{1}]]_{+} → [[A, D_{2}]]_{+} or the map Ẽ_{A}(D_{1}) → Ẽ_{A}(D_{2}) induced by a (full) homomorphism φ : D_{1} → D_{2}.

3.1.6 Lemma. Let A be separable, nuclear, unital, and simple. Let

0 ⟶ J ⟶^{μ} D ⟶^{π} D/J ⟶ 0

be a short exact sequence of separable C^{∗}-algebras. Then the sequence

Ẽ_{A}(J) ⟶^{μ_{∗}} Ẽ_{A}(D) ⟶^{π_{∗}} Ẽ_{A}(D/J)

is exact in the middle.

Proof: It is immediate that π_{∗} ∘ μ_{∗} = 0.

For the other half, we introduce the maps χ_{D} : D^{#} → K ⊗ O_{∞} and ι_{D} : K ⊗ O_{∞} → D^{#} associated with the unitization maps D^{+} → C and C → D^{+}. Define χ_{D/J}, ι_{D/J}, etc. similarly. Now let η ∈ ker(π_{∗}), and choose a full asymptotic morphism φ : A → D^{#} whose class is η. By definition, we have [[π^{#} ∘ φ]] = 0 in [[A, (D/J)^{#}]]_{+}. Choose a full homomorphism ζ : A → K ⊗ O_{∞} with a standard factorization through O_{2} ⊗ A, as in the proof of Proposition 3.1.2. Theorem 2.3.7 then implies that π^{#} ∘ φ is asymptotically unitarily equivalent to ι_{D/J} ∘ζ, so there is a unitary path t → u_{t} in ((D/J)^{#})^{+} such that u_{t}(π^{#} ∘ φ_{t})(a)u_{t}^{∗} → (ι_{D/J} ∘ ζ)(a) for all a ∈ A.

Without changing homotopy classes, we may replace φ by φ ⊕ 0 and ζ by ζ ⊕ 0. This also replaces π^{#} ∘ φ and ι_{D/J} ∘ ζ by their direct sums with the zero asymptotic morphism. We then replace u_{t} by u_{t} ⊕ u_{t}^{∗}. We may thus assume without loss of generality that u is in the identity component of the unitary group of C_{b}([0, ∞), ((D/J)^{#})^{+}). Therefore there is v ∈ U_{0}(C_{b}([0, ∞), (D^{#})^{+})) whose image is u. Then π^{#}(v_{t}) = u_{t} for all t, whence

lim_{t→0} π^{#}(v_{t}φ_{t}(a)v_{t}^{∗} − (ι_{D} ∘ ζ)(a)) = 0

for all a ∈ A.

Let σ : (D/J)^{#} → D^{#} be a continuous (nonlinear) cross section for π^{#} satisfying σ(0) = 0. (See [1].) Define ψ_{t} : A → D^{#} by

ψ_{t}(a) = v_{t}φ_{t}(a)v_{t}^{∗} − (σ ∘ π^{#}) (v_{t}φ_{t}(a)v_{t}^{∗} − (ι_{D} ∘ ζ)(a)) .

This yields an asymptotic morphism asymptotically equal to t ↦ v_{t}φ_{t}(−)v_{t}^{∗}, and hence asymptotically unitarily equivalent to φ. Furthermore, π^{#}(ψ_{t}(a) − (ι_{D} ∘ ζ)(a)) = 0 for all t and a. It follows that ψ_{t}(a) ∈ J^{#} and that χ_{J}(ψ_{t}(a)) = ζ(a). So ψ is in fact a full asymptotic morphism from A to J^{#} such that [[χ_{J} ∘ ψ]] = 0, from which it follows that ψ defines a class [[ψ]] ∈ Ẽ_{A}(J). Clearly μ_{∗}([[ψ]]) = η. This shows that ker(π_{∗}) ⊂ Im(μ_{∗}).

3.1.7 Corollary. Let A be separable, nuclear, unital, and simple. Let

0 ⟶ J ⟶^{μ} D ⟶^{π} D/J ⟶ 0

be a split short exact sequence of separable C^{∗}-algebras. Then there is a natural split exact sequence

0⟶Ẽ_{A}(J) ⟶^{μ_{∗}} Ẽ_{A}(D) ⟶^{π_{∗}} Ẽ_{A}(D/J) ⟶ 0.

Proof: This is Proposition 4.1 (b) of [15], noting that the proof of Part (b) there doesn’t use stability. Indeed, since Ẽ_{A} is middle exact (the previous lemma) and homotopy invariant, Lemma 5 in Section 7 of [26] provides a long exact sequence

⋯ ⟶^{(Sμ)_{∗}} Ẽ_{A}(SD) ⟶^{(Sπ)_{∗}} Ẽ_{A}(S(D/J))⟶Ẽ_{A}(J) ⟶^{μ_{∗}} Ẽ_{A}(D) ⟶^{π_{∗}} Ẽ_{A}(D/J).

The desired conclusion can now be immediately obtained using the splitting map.

3.1.8 Remark. It should be pointed out that we need much less than the full strength of Theorem 2.3.7 here. Only knowing that homotopy implies asymptotic unitary equivalence for full asymptotic morphisms from A to K ⊗ O_{∞} ⊗ C([0, 1]), it is possible to prove middle exactness in the first stage of the Puppe sequence, namely

Ẽ_{A}(Cπ) ⟶ Ẽ_{A}(D) ⟶ Ẽ_{A}(D/J).

This sequence can be extended to the left as in the proof of Proposition 2.6 of [56]. Proposition 3.2 of [16] can then be used to show that Ẽ_{A} is split exact.

We now prove stability of Ẽ_{A} under formation of tensor products with both K and O_{∞}.

3.1.9 Lemma. Let A be separable, nuclear, unital, and simple, and let D be a separable C^{∗}-algebra. Then the map d ↦ 1 ⊗ d, from D to O_{∞} ⊗ D, induces an isomorphism Ẽ_{A}(D) → Ẽ_{A}(O_{∞} ⊗ D).

Proof: By naturality, Proposition 3.1.7, and the Five Lemma, it suffices to prove this for unital D. By Remark 3.1.5, we have to prove that d ↦ 1 ⊗ d induces an isomorphism [[A, K ⊗ O_{∞} ⊗ D]]_{+} → [[A, K ⊗ O_{∞} ⊗ O_{∞} ⊗ D]]_{+}. This follows from Theorem 2.1.5 and Proposition 2.1.11, since these results imply that the map x ↦ x ⊗ 1, from O_{∞} to O_{∞} ⊗ O_{∞}, is homotopic to an isomorphism.

The other stability result requires the following lemma. We really want an increasing continuously parametrized approximate identity of projections, but of course such a thing does not exist. The quasiincreasing version in the lemma is good enough.

3.1.10 Lemma. Let D be a unital purely infinite simple C^{∗}-algebra, and let e_{0} ∈ K ⊗ D be a nonzero projection. Then there exists a continuous family t ↦ e_{t} of projections in K ⊗ D such that, for every b ∈ K ⊗ D, we have

lim_{t→∞}(e_{t}b − b) = lim_{t→∞}(be_{t} − b) = lim_{t→∞}(e_{t}be_{t} − b) = 0,

such that e_{0} is the given projection, and such that e_{s} ≥ e_{t} for s ≥ t + 1.

Proof: Choose a nonzero projection p ∈ K ⊗ D such that [p] = 0 in K_{0}(D). We start by constructing a family t ↦ f_{t} in K ⊗ pDp. Note that

[diag(1_{pDp}, 0, 0)] = [diag(1_{pDp}, 1_{pDp}, 0)] = 0

in K_{0}(M_{3}(pDp)). Therefore there is a homotopy t ↦ q_{t} of projections in M_{3}(pDp) such that

q_{0} = diag(1, 0, 0) and q_{1} = diag(1, 1, 0).

Now define

f_{n+s} = 1_{M_{n+1}(pDp)} ⊕ q_{s} ⊕ 0 ∈ K ⊗ pDp

for n = 0, 1, … and s ∈ [0, 1]. The family f_{t} is clearly continuous. It satisfies f_{0} = p⊕p. We have f_{t} ≥ 1_{M_{n+1}(pDp)} for t ≥ n, so t ↦ f_{t} really is an approximate identity. Finally, f_{t} ≤ 1_{M_{n+3}(pDp)} for t ≤ n, so f_{s} ≥ f_{t} for s ≥ t + 4. We can replace 4 by 1 in this last statement by a reparametrization.

To get the general case, choose a projection r ∈ pDp with [r] = −[e_{0}] in K_{0}(D). Then f_{t} ≥ p ≥ r for all t, so t ↦ f_{t} − r is a continuously parametrized approximate identity of projections for (1 − r)(K ⊗ pDp)(1 − r). (Here 1 is the identity of (K ⊗ pDp)^{+}.) There is an isomorphism

φ : K ⊗ D → (1 − r)(K ⊗ pDp)(1 − r),

and since [f_{0} − r] = [e_{0}] in K_{0}(D), we can require that φ(e_{0}) = f_{0} − r. Now set e_{t} = φ^{−1}(f_{t} − r). Then clearly e_{t}b − b, be_{t} − b → 0 as t → ∞. It follows that

∥e_{t}be_{t} − b∥ ≤ ∥e_{t}b − b∥ ∥e_{t}∥ + ∥be_{t} − b∥ → 0

as well. 3.1.11 Lemma. Let A be separable, nuclear, unital, and simple, let D be separable, and let e ∈ K be a rank one projection. Then the map d ↦ e ⊗ d, from D to K ⊗ D, induces an isomorphism Ẽ_{A}(D) → Ẽ_{A}(K ⊗ D).

Proof: By Lemma 3.1.9, we may use O_{∞} ⊗ D in place of D, and as in its proof we may assume D is unital.

Let s ∈ O_{∞} be a proper isometry, and define γ : O_{∞} ⊗ D → O_{∞} ⊗ D by γ(a) = (s ⊗ 1)a(s ⊗ 1)^{∗}. We claim that γ_{∗} : Ẽ_{A}(O_{∞} ⊗ D) → Ẽ_{A}(O_{∞} ⊗ D) is an isomorphism. It follows from Remark 3.1.5 and Definition 3.1.3 that this map can be thought of as composition with id_{K⊗O_{∞}} ⊗ γ from [[A, K ⊗ O_{∞} ⊗ O_{∞} ⊗ D]]_{+} to itself, even though γ is not unital. (The discrepancy is an orthogonal sum with an asymptotic morphism which up to homotopy has a trivializing factorization. Note that the composition with γ is still full.) Now K ⊗ ss^{∗}O_{∞}ss^{∗} and K ⊗ (1 − ss^{∗})O_{∞}(1 − ss^{∗}) are both isomorphic to K ⊗ O_{∞}, so we may as well consider the map from [[A, K ⊗ O_{∞} ⊗ O_{∞} ⊗ D]]_{+} to [[A, M_{2}(K ⊗ O_{∞} ⊗ O_{∞} ⊗ D)]]_{+} induced by inclusion in the upper right corner. Let τ : M_{2}(K) → K be an isomorphism. Then a ↦ τ(a ⊕ 0) is homotopic to id_{K} and b ↦ τ(b) ⊕ 0 is homotopic to id_{M_{2}(K)}. So our map has an inverse given by composition with τ ⊗ id_{O_{∞}⊗O_{∞}⊗D}.

We next require a construction involving O_{∞} and K ⊗ O_{∞}. Define φ : O_{∞} → K ⊗ O_{∞} by φ(x) = e ⊗ x. Let t ↦ e_{t} be a continuously parametrized approximate identity for K ⊗ O_{∞} which satisfies the properties of the previous lemma and has e_{0} = e ⊗ 1. Let t ↦ u_{t} be a continuous family of unitaries in (K ⊗ O_{∞})^{+} such that u_{0} = 1 and u_{t}e_{t}u_{t}^{∗} = e_{0} for all t. Define ψ_{t}^{(0)} : K ⊗ O_{∞} → K ⊗ O_{∞} by ψ_{t}^{(0)}(a) = u_{t}e_{t}ae_{t}u_{t}^{∗}. One immediately checks that ψ^{(0)} is an asymptotic morphism whose values are in (e ⊗ 1)(K ⊗ O_{∞})(e ⊗ 1), so that there is an asymptotic morphism t ↦ ψ_{t} from K ⊗ O_{∞} to O_{∞} such that φ ∘ ψ_{t} = ψ_{t}^{(0)} for all t.

The composite asymptotic morphisms φ ∘ ψ and ψ ∘ φ can be computed without reparametrization, because φ is a homomorphism. Now φ ∘ ψ = ψ^{(0)}, which is asymptotically unitarily equivalent to (t, a) ↦ e_{t}ae_{t}, which in turn is asymptotically equal to id_{K⊗O_{∞}}. So φ ∘ ψ is homotopic to id_{K⊗O_{∞}}. Also, ψ ∘ φ is clearly homotopic to a map of the form x ↦ sxs^{∗} for a proper isometry s ∈ O_{∞}.

We now observe that id_{K⊗O_{∞}} ⊗ (φ ⊗ id_{D})^{+} and id_{K⊗O_{∞}} ⊗ (ψ ⊗ id_{D})^{+} define full asymptotic morphisms from (O_{∞} ⊗ D)^{#} to (K ⊗ O_{∞} ⊗ D)^{#} and back. The composite from (K ⊗O_{∞}⊗D)^{#} to itself is homotopic to the identity, and therefore induces the identity map on Ẽ_{A}(K ⊗ O_{∞} ⊗ D). Composition on the right with the composite from (O_{∞} ⊗ D)^{#} to itself is a map of the form γ_{∗} as considered at the beginning of the proof, and is thus an isomorphism from Ẽ_{A}(O_{∞} ⊗ D) to itself. It follows that φ_{∗} is an isomorphism.

3.2 The isomorphism with KK-theory

In this subsection, we prove that if A is separable, nuclear, unital, and simple, and D is separable, then the natural map from Ẽ_{A}(D) to KK^{0}(A, D) is an isomorphism. Combined with Remark 3.1.5, this gives for unital D a form of “unsuspended E-theory” as in [16], in which we need only discard the zero asymptotic morphism.

We will use the universal property of KK-theory with respect to split exact, stable, and homotopy invariant functors on separable C^{∗}-algebras [24]. (We use this instead of the related property of E-theory because it is more convenient for the proof of Lemma 3.2.3 below.)

3.2.1 Notation. In this subsection, we denote by S the category of separable C^{∗}-algebras and homomorphisms and by KK the category of separable C^{∗}-algebras with morphisms KK^{0}(A, B) for C^{∗}-algebras A and B. If η ∈ KK^{0}(A, B) and λ ∈ KK^{0}(B, C), we denote their product by λ × η ∈ KK^{0}(A, C). We further denote by k the functor from S to KK which sends a homomorphism to the class it defines in KK-theory.

3.2.2 Lemma. Let A be separable, nuclear, unital, and simple. Then there is a functor Ê_{A} from KK to the category of abelian groups such that Ê_{A} ∘ k = Ẽ_{A}.

This simply means that one can make sense of Ẽ_{A}(η) : Ẽ_{A}(D) → Ẽ_{A}(F) not only when η is an asymptotic morphism from D to F, but also when η is merely an element of KK^{0}(D, F).

Proof of Lemma 3.2.2: The result is immediate from Theorem 4.5 of [24], since Ẽ_{A} is a stable (Lemma 3.1.11), split exact (Corollary 3.1.7), and homotopy invariant (Proposition 3.1.4) functor from separable C^{∗}-algebras to abelian groups.

We want to show that Ẽ_{A}(D) is naturally isomorphic to KK^{0}(A, D). Our argument is based on an alternate proof of the main theorem of [16] suggested by the referee of that paper; we are grateful to Marius Dǎdǎrlat for telling us about it. The argument requires the construction of certain natural transformations. (The argument used in Section 4 of [16] presumably also works.)

Before starting the construction, we prove a lemma on the functors F̂ of Higson [24] (as used in the previous lemma).

3.2.3 Lemma. Let F and G be stable, split exact, and homotopy invariant functors from S to the category of abelian groups, and let F̂ and Ĝ be the unique extensions to functors from KK of Theorem 4.5 of [24]. (In particular, F or G could be Ẽ_{A} and F̂ or Ĝ could be Ê_{A}, as in Lemma 3.2.2.) If α is a natural transformation from F to G, then α is also a natural transformation from F̂ to Ĝ.

Proof: Let μ ∈ KK^{0}(A, B). By Lemma 3.6 of [24], we can choose a representative cycle (in the sense of Definition 2.1 of [24]) of the form Φ = (φ_{+}, φ_{−}, 1), where φ_{+}, φ_{−} : A → M(K ⊗B) are homomorphisms such that φ_{+}(a)−φ_{−}(a) ∈ K ⊗ B for a ∈ A. The homomorphism F̂(μ) is then the composite

F (A) ⟶^{F (_{φ_{+}})−F (_{φ_{−}})} F (A_{Φ}) ⟶^{F (π)} F (K ⊗ B) ⟶^{F (ε)^{−1}} F (B),

for a certain C^{∗}-algebra A_{Φ}, certain homomorphisms π, φ̂_{+}, and φ̂_{−}, and with ε(a) = 1 ⊗ a. (See Definition 3.4 and the proofs of Theorems 3.7 and 4.5 in [24].) From this expression, it is obvious that naturality with respect to homomorphisms implies naturality with respect to classes in KK-theory.

3.2.4 Definition. Let A be separable, nuclear, unital, and simple. We regard KK^{0}(A, −) and Ê_{A} as functors from KK to abelian groups. (On morphisms, the first of these sends η ∈ KK^{0}(D_{1}, D_{2}) to Kasparov product with η.) We now define natural transformations

α : KK^{0}(A, −) → Ê_{A} and β : Ê_{A} → KK^{0}(A, −).

To define α_{D}, let e ∈ K be a rank one projection, let ι_{A} : A → K ⊗O_{∞} ⊗A be the map ι_{A}(a) = e ⊗ 1 ⊗ a, and let [[ι_{A}]] ∈ Ẽ_{A}(A) denote its class in [[A, K ⊗ O_{∞} ⊗ A]]_{+} ≅ Ẽ_{A}(A). (Recall that A is unital, so that Remark 3.1.5 applies.) Now let η ∈ KK^{0}(A, D). Then Ê_{A}(η) is a homomorphism from Ẽ_{A}(A) to Ẽ_{A}(D). Define

α_{D}(η) = Ê_{A}(η)([[ι_{A}]]) ∈ Ẽ_{A}(D).

To define β_{D}, let χ_{D} : D^{#} → K ⊗O_{∞} be the standard map (as in the proof of Lemma 3.1.6). Starting with η ∈ Ẽ_{A}(D) ⊂ [[A, D^{#}]], choose a full asymptotic morphism φ : A → D^{#} with [[χ_{D}]] · [[φ]] = 0 which represents η. Now recall (Corollary 9 (b) of [11]; Section 5 of [54]) that for A and B separable and A (K-)nuclear, there is a canonical isomorphism E(A, B) ≅ KK^{0}(A, B). Further recall that there is a canonical isomorphism KK^{0}(SA, SB) ≅ KK^{0}(A, B) (Theorem 7 of Section 5 of [26]). Form the second suspension

[[S^{2}φ]] ∈ [[S^{2}A, S^{2}D^{#}]] = E(SA, SD^{#})
≅ KK^{0}(SA, SD^{#}) ≅ KK^{0}(A, O_{∞} ⊗ D^{+}),

and regard [[S^{2}φ]] as an element of KK^{0}(A, O_{∞}⊗D^{+}). Since [[S^{2}χ_{D}]]·[[S^{2}φ]] = 0, split exactness of KK^{0}(A, −) implies that [[S^{2}φ]] is actually in KK^{0}(A, O_{∞}⊗ D). In this last expression, we can use the KK-equivalence of O_{∞} and C, given by the unital homomorphism C → O_{∞}, to drop O_{∞}. We thus obtain an element β_{D}(η) ∈ KK^{0}(A, D).

3.2.5 Lemma. The maps α_{D} and β_{D} of the previous definition are in fact natural transformations.

Proof: It is easy to check that both α and β are natural with respect to homomorphisms, so naturality with respect to classes in KK-theory follows from Lemma 3.2.3.

3.2.6 Theorem. Let A be separable, nuclear, unital, and simple. Then for every separable D, the maps α_{D} and β_{D} of Definition 3.2.4 are mutually inverse isomorphisms.

Proof: It is convenient to prove this first under the assumptions that O_{∞} ⊗A ≅ A and O_{∞} ⊗ D ≅ D. It then follows that the map a ↦ 1 ⊗ a from A to O_{∞} ⊗ A is homotopic to an isomorphism, and similarly for D. (This is true for O_{∞} by Theorem 2.1.5 and Proposition 2.1.11. Therefore it is true for O_{∞} ⊗ A and O_{∞} ⊗ D, hence for A and D.) Thus, A and O_{∞} ⊗ A are naturally homotopy equivalent, and therefore also naturally equivalent in KK as well.

Similar considerations apply to D. Thus, Ẽ_{A}(D) becomes just [[A, K ⊗ D]]_{+}. The natural transformations above are then given by

α_{D}(η) = Ê_{A}(η)([[id_{A}]])

(with id_{A} being the obvious map from A to K ⊗ A), and

β_{D}([[φ]]) = [[S^{2}φ]] ∈ [[S^{2}A, K ⊗ S^{2}D]] ≅ KK^{0}(A, D).

Letting 1_{A} denote the class in KK^{0}(A, A) of the identity map, we then immediately verify that

α_{A}(1_{A}) = [[id_{A}]] and β_{A}([[id_{A}]]) = 1_{A}.

We now show that these two facts imply the theorem for unital D. Let η ∈ KK^{0}(A, D). Then η = 1_{A} × η, and naturality implies that

β_{D}(α_{D}(1_{A} × η)) = β_{D}(Ê_{A}(η)(α_{A}(1_{A}))) = β_{A}(α_{A}(1_{A})) × η = 1_{A} × η.

So β_{D} ∘ α_{D} = id. For the other direction, let μ ∈ Ê_{A}(D). Using Corollary 2.3.10, represent μ as the class of a full homomorphism φ : A → K ⊗ D. Let η = [[S^{2}φ]] be the KK-class determined by [[φ]]. Then, identifying K ⊗K with K as necessary, we have

μ = φ_{∗}([[id_{A}]]) = Ê_{A}(η)([[id_{A}]]).

The same argument as above now shows that

(α_{D} ∘ β_{D}) (Ê_{A}(η)([[id_{A}]])) = Ê_{A}(η)([[id_{A}]]).

So α_{D} ∘ β_{D} = id also.

The result for nonunital algebras follows from naturality, split exactness, and the Five Lemma.

To remove the assumption that O_{∞} ⊗ D ≅ D, use Lemma 3.1.9.

Finally, we remove the assumption that O_{∞}⊗A ≅ A. Let δ_{0} : O_{∞}⊗O_{∞} → O_{∞} be an isomorphism (from Theorem 2.1.5), and let δ : O_{∞} ⊗ K ⊗ O_{∞} → K ⊗ O_{∞} be the obvious corresponding map. Define i_{D} : Ẽ_{A}(D) → Ẽ_{O_{∞}⊗A}(D) by

i_{D}([[η]]) = [[δ ⊗ id_{D^{+}}]] · [[id_{O_{∞}} ⊗ η]].

Let j_{D} : KK^{0}(A, D) → KK^{0}(O_{∞} ⊗ A, D) be the isomorphism induced by the KK-equivalence of C and O_{∞}. Both i and j are natural transformations. Using Theorem 2.1.5 and Proposition 2.1.11, we can rewrite j_{O_{∞}⊗D}(μ) as (δ_{0} ⊗ id_{D})_{∗}(1_{O_{∞}} ⊗ μ). This formula and Remark 3.1.5 imply that i_{D} ∘ α_{D} = α_{D} ∘ j_{D} when D is unital and O_{∞} ⊗ D ≅ D. The previous paragraph and the definition of Ẽ_{A}(D) in terms of [[A, D^{#}]]_{+} now imply that i_{D} ∘ α_{D} = α_{D} ∘ j_{D} for all D. A related argument shows that also j_{D} ∘ β_{D} = β_{D} ∘ i_{D} for all D.

It now suffices to prove that i_{D} is an isomorphism for all D. By naturality, split exactness, and the Five Lemma, it suffices to do so for unital D. In this case, we have

i_{D} : [[A, K ⊗ O_{∞} ⊗ D]]_{+} → [[O_{∞} ⊗ A, K ⊗ O_{∞} ⊗ D]]_{+}

given by i_{D}([[η]]) = [[δ ⊗ id_{D}]] · [[id_{O_{∞}} ⊗ η]]. Define a map k_{D} in the opposite direction by restriction to 1 ⊗ A ⊂ O_{∞} ⊗ A. We prove that k_{D} = i_{D}^{−1}.

Let δ̃(x) = δ(1 ⊗ x). Proposition 2.1.11 implies that there is a homotopy δ̃ ≃ id_{K⊗O_{∞}}. It is easy to check directly that k_{D}∘i_{D} sends [[η]] to [[(δ̃⊗id_{D})∘η]], so k_{D} ∘ i_{D} is the identity. For the reverse composition, let τ_{A} be the inclusion of A = 1 ⊗ A in O_{∞} ⊗ A, and let φ : O_{∞} ⊗ O_{∞} → O_{∞} ⊗ O_{∞} be the flip φ(x ⊗ y) = y ⊗ x. Then φ ≃ id_{O_{∞}⊗O_{∞}} by Proposition 2.1.11 and Theorem 2.1.5. Therefore, for [[η]] ∈ [[O_{∞} ⊗ A, K ⊗ O_{∞} ⊗ D]]_{+}, we have

(δ ⊗ id_{D}) ∘ (id_{O_{∞}} ⊗ (η ∘ τ_{A}))

≃ (δ ⊗ id_{D}) ∘ (id_{O_{∞}} ⊗ η) ∘ (φ ⊗ id_{A}) ∘ (id_{O_{∞}} ⊗ τ_{A}) = (δ̃ ⊗ id_{D}) ∘ η ≃ η.

This shows that i_{D} ∘ k_{D} is the identity.

3.2.7 Remark. We used Corollary 2.3.10 in this proof because we had it available. It is, however, not necessary for the argument. Using methods similar to, but a bit more complicated than, the proof of Lemma 3.2.3, one can show that if F as there is in fact a functor on homotopy classes of asymptotic morphisms, then F ([[φ]]) is equal to F̂ applied to the KK-theory class given by φ.

3.2.8 Theorem. Let A be a separable unital nuclear simple C^{∗}-algebra. Then for separable unital C^{∗}-algebras D, the set of homotopy classes of full asymptotic morphisms from A to K ⊗O_{∞} ⊗D is naturally isomorphic to KK^{0}(A, D) via the map sending an asymptotic morphism to the KK-class it determines.

Proof: This follows from Theorem 3.2.6 and Remark 3.1.5.

4 Theorems on KK-theory and classification

In this section, we present our main results. The first subsection contains the alternate descriptions of KK-theory in terms of homotopy classes and asymptotic unitary equivalence classes of homomorphisms, in case the first variable is separable, nuclear, unital, and simple. We also give here a proof that homotopies of automorphisms of separable nuclear unital purely infinite simple C^{∗}- algebras can in fact be chosen to be isotopies. The second subsection contains the classification theorem and its corollaries. The third subsection contains the nonclassification results.

4.1 Descriptions of KK-theory

Probably the most striking of our descriptions of KK-theory is the following:

4.1.1 Theorem. For a separable unital nuclear simple C^{∗}-algebra A and a separable unital C^{∗}-algebra D, the obvious maps define natural isomorphisms of abelian groups between the following three objects:

(1) The set of asymptotic unitary equivalence classes of full homomorphisms from A to K⊗O_{∞}⊗D, with the operation given by direct sum (Definition 1.1.3).

(2) The set of homotopy classes of full homomorphisms from A to K ⊗ O_{∞} ⊗ D, with the operation given by direct sum as above.

(3) The group KK^{0}(A, D).

Proof: For the purposes of this proof, denote the set in (1) by KU(A, D) and the set in (2) by KH(A, D). The map from KH(A, D) to KK^{0}(A, D) is the one from Theorem 3.2.8. By this theorem, we can use [[A, K ⊗ O_{∞} ⊗ D]]_{+} in place of KK^{0}(A, D).

Lemma 1.3.3 (2) implies that the map from KU(A, D) to KH(A, D) is well defined, and it is then clearly surjective. Injectivity is immediate from Theorem 2.3.7. Thus this map is an isomorphism. Theorem 3.2.8 implies that the map from KH(A, D) to [[A, K⊗O_{∞}⊗D]]_{+} is injective, while Corollary 2.3.10 implies that the map from KU(A, D) to [[A, K ⊗ O_{∞} ⊗ D]]_{+} is surjective. Therefore these maps are in fact both isomorphisms. It now follows that KU(A, D) and KH(A, D) are both abelian groups.

We now want to give a stable version of this theorem, in which the Kasparov product will reduce exactly to composition of homomorphisms. We need the following lemma. The hypotheses allow one continuous path of homomorphisms, and require unitaries in U_{0}((K ⊗ D)^{+}), for use in the next subsection.

4.1.2 Lemma. Let A be separable, nuclear, unital, and simple, let D_{0} be separable and unital, and let D = O_{∞} ⊗ D_{0}. Let t ↦ φ_{t}, for t ∈ [0, ∞), be a continuous path of full homomorphisms from K ⊗ A to K ⊗ D, and let ψ : K ⊗ A → K ⊗ D be a full homomorphism. Assume that [φ_{0}] = [ψ] in KK^{0}(A, D). Then there is an asymptotic unitary equivalence from φ to ψ which consists of unitaries in U_{0}((K ⊗ D)^{+}).

Proof: Let {e_{ij}} be a system of matrix units for K. Identify A with the subalgebra e_{11} ⊗ A of K ⊗ A. Define φ_{t}^{(0)} and ψ^{(0)} to be the restrictions of φ_{t} and ψ to A. Then [φ_{0}^{(0)}] = [ψ^{(0)}] in KK^{0}(A, D). It follows from Theorem 4.1.1 that φ_{0}^{(0)} is homotopic to ψ^{(0)}. Therefore φ^{(0)} and ψ^{(0)} are homotopic as asymptotic morphisms, and Theorem 2.3.7 provides an asymptotic unitary equivalence t ↦ u_{t} in U((K ⊗ D)^{+}) from φ^{(0)} to ψ^{(0)}. Let c ∈ U((K ⊗ D)^{+}) be a unitary with cψ^{(0)}(1) = ψ^{(0)}(1)c = ψ^{(0)}(1) and such that c is homotopic to u_{0}^{−1}. Then c commutes with every ψ^{(0)}(a). Replacing u_{t} by cu_{t}, we obtain an asymptotic unitary equivalence, which we again call t ↦ u_{t}, from φ^{(0)} to ψ^{(0)} which is in U_{0}((K ⊗ D)^{+}).

Define \overline{e}_{ij} = e_{ij} ⊗ 1. Then in particular u_{t}φ_{t}(\overline{e}_{11})u_{t}^{∗} → ψ(\overline{e}_{11}) as t → ∞. Therefore there is a continuous path t → z_{t}^{(1)} ∈ U_{0}((K ⊗ D)^{+}) such that z_{t}^{(1)} → 1 and z_{t}^{(1)}u_{t}φ_{t}(\overline{e}_{11})u_{t}^{∗}(z_{t}^{(1)})^{∗} = ψ(\overline{e}_{11}) for all t. We still have z_{t}^{(1)}u_{t}φ_{t}(e_{11} ⊗ a)u_{t}^{∗}(z_{t}^{(1)})^{∗} → ψ(e_{11} ⊗ a) for a ∈ A.

For convenience, set f_{ijt} = z_{t}^{(1)}u_{t}φ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗}, for all t and for 1 ≤ i, j ≤ 2. For each fixed t, the f_{ijt} are matrix units, and f_{11t} = ψ(\overline{e}_{11}). Set w_{t} = ψ(\overline{e}_{21})f_{12t} + 1 − f_{22t} ∈ U((K ⊗ D)^{+}). Then one checks that w_{t}f_{ijt}w_{t}^{∗} = ψ(\overline{e}_{ij}) for all t and for 1 ≤ i, j ≤ 2. Choose c ∈ U((K ⊗ D)^{+}) with

cψ(\overline{e}_{11} + \overline{e}_{22}) = ψ(\overline{e}_{11} + \overline{e}_{22})c = ψ(\overline{e}_{11} + \overline{e}_{22}) and cw_{1} ∈ U_{0}((K ⊗ D)^{+}).

Set z_{t}^{(2)} = cw_{t} for t ≥ 1 and extend z_{t}^{(2)} over [0, 1] to be continuous, unitary, and satisfy z_{0}^{(2)} = 1. This gives z_{t}^{(2)} = 1 for t = 0, z_{t}^{(2)}ψ(\overline{e}_{11}) = ψ(\overline{e}_{11})z_{t}^{(2)} = ψ(\overline{e}_{11}) for all t, and

z_{t}^{(2)} [z_{t}^{(1)}u_{t}φ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗}] (z_{t}^{(2)})^{∗} = ψ(\overline{e}_{ij})

for t ≥ 1 and 1 ≤ i, j ≤ 2.

We continue inductively, obtaining by the same method a sequence of continuous paths t ↦ z_{t}^{(n)} such that z_{t}^{(n+1)} = 1 for t ≤ n − 1,

z_{t}^{(n+1)} (∑_{j=1}^{n} ψ(\overline{e}_{jj})) = (∑_{j=1}^{n} ψ(\overline{e}_{jj})) z_{t}^{(n+1)} = ∑_{j=1}^{n} ψ(\overline{e}_{jj})

for all t, and

z_{t}^{(n+1)} [(z_{t}^{(1)}z_{t}^{(2)} ⋯ z_{t}^{(n)}) u_{t}φ_{t}(\overline{e}_{ij})u_{t}^{∗} (z_{t}^{(1)}z_{t}^{(2)} ⋯ z_{t}^{(n)})^{∗}] (z_{t}^{(n+1)})^{∗} = ψ(\overline{e}_{ij})

for t ≥ n and 1 ≤ i, j ≤ n + 1.

Now define

z_{t} = ( lim_{n→∞} z_{t}^{(1)}z_{t}^{(2)} ⋯ z_{t}^{(n)}) u_{t}.

In a neighborhood of each t, all but finitely many of the z_{t}^{(k)} are equal to 1, so this limit of products yields a continuous path of unitaries in U_{0}((K ⊗ D)^{+}). Moreover, z_{t}φ_{t}(\overline{e}_{ij})z_{t}^{∗} = ψ(\overline{e}_{ij}) whenever t ≥ i, j, so that lim_{t→∞} z_{t}φ_{t}(\overline{e}_{ij})z_{t}^{∗} = ψ(\overline{e}_{ij}) for all i and j, while

lim_{t→∞} z_{t}φ_{t}(e_{11} ⊗ a)z_{t}^{∗} = lim_{t→∞} z_{t}^{(1)}u_{t}φ_{t}(e_{11} ⊗ a)u_{t}^{∗}(z_{t}^{(1)})^{∗} = ψ(e_{11} ⊗ a)

for all a ∈ A. Since the \overline{e}_{ij} and e_{11} ⊗ a generate K ⊗ A, this shows that t ↦ z_{t} is an asymptotic unitary equivalence.

4.1.3 Theorem. For a separable unital nuclear simple C^{∗}-algebra A and a separable unital C^{∗}-algebra D, the obvious maps and the isomorphism KK^{0}(A, D) → KK^{0}(K ⊗ O_{∞} ⊗ A, K ⊗ O_{∞} ⊗ D) define natural isomorphisms of abelian groups between the following three objects:

(1) The set of asymptotic unitary equivalence classes of full homomorphisms from K ⊗ O_{∞} ⊗ A to K ⊗ O_{∞} ⊗ D, with the operation given by direct sum (as in Theorem 4.1.1).

(2) The set of homotopy classes of full homomorphisms from K ⊗ O_{∞} ⊗ A to K ⊗ O_{∞} ⊗ D, with the operation given by direct sum as above.

(3) The group KK^{0}(A, D).

Moreover, if B is another a separable unital nuclear simple C^{∗}-algebra, then the Kasparov product KK^{0}(A, B) × KK^{0}(B, D) → KK^{0}(A, D) is given in the groups in (1) and (2) by composition of homomorphisms.

Proof: The last statement will follow immediately from the rest of the theorem, since if two KK-classes are represented by homomorphisms, then their product is represented by the composition.

For the rest of the theorem, first note that the map KK^{0}(A, D) → KK^{0}(K ⊗O_{∞} ⊗A, K ⊗O_{∞} ⊗D) is a natural isomorphism because it is induced by the KK-equivalence C → K ⊗ O_{∞}, given by 1 ↦ e ⊗ 1 for some rank one projection e ∈ K, in each variable.

Now observe that the previous lemma implies that the map from the set in (1) to KK^{0}(A, D) is injective. Moreover, the map from the set in (1) to the set in (2) is well defined by Lemma 1.3.3 (2), and is then obviously surjective. It therefore suffices to prove that the map from the set in (2) to KK^{0}(A, D) is surjective, that is, that every class in KK^{0}(A, D) is represented by a homomorphism from K ⊗ O_{∞} ⊗ A to K ⊗ O_{∞} ⊗ D. It follows from Theorem 4.1.1 that every such class is represented by a homomorphism from A to K ⊗O_{∞}⊗D, and we obtain a homomorphism from K ⊗O_{∞}⊗A to K ⊗O_{∞}⊗D by tensoring with id_{K⊗O_{∞}} and composing with the tensor product of id_{D} and an isomorphism K ⊗ O_{∞} ⊗ K ⊗ O_{∞} → K ⊗ O_{∞} which is the identity on K-theory.

We finish this section with one other application. Following terminology from differential topology, we define an isotopy to be a homotopy t ↦ φ_{t} in which each φ_{t} is an isomorphism.

4.1.4 Theorem. Let A be a separable nuclear unital purely infinite simple C^{∗}-algebra.

(1) If U(A) is connected, then two automorphisms of A with the same class in KK^{0}(A, A) are isotopic.

(2) Any two automorphisms of K ⊗ A with the same class in KK^{0}(A, A) are isotopic.

Proof: For (2), take D = A in Lemma 4.1.2, note that O_{∞} ⊗ A ≅ A (Theorem 2.1.5), and note that an asymptotic unitary equivalence with unitaries in U_{0}((K ⊗ A)^{+}) gives an isotopy, not just a homotopy.

For (1), let φ and ψ be automorphisms of A with the same class in KK^{0}(A, A). Let e ∈ K be a rank one projection. Apply (2) to id_{K} ⊗ φ and id_{K} ⊗ ψ. Thus, there is a unitary path t ↦ u_{t} in (K ⊗ A)^{+} with u_{t}φ(e ⊗ a)u_{t}^{∗} → ψ(e ⊗ a) for a ∈ A. In particular, u_{t}(e ⊗ 1)u_{t}^{∗} → (e ⊗ 1). Replacing u_{t} by v_{t}u_{t} for a suitable unitary path t ↦ v_{t}, we may therefore assume that u_{t}(e ⊗ 1)u_{t}^{∗} = e ⊗ 1 for all t. Cut down by e ⊗ 1, and observe that the hypotheses imply that (e ⊗ 1)u_{0}(e ⊗ 1) is homotopic to 1. Now finish as in the proof of (2).

4.2 The classification theorem

The following theorem is the stable version of the main classification theorem. Everything else will be an essentially immediate corollary.

In the proof, it is easy to get the existence of the isomorphism; this is just the by now well known Elliott approximate intertwining argument. We need a more complicated version of this argument to make sure that the isomorphism we construct has the right class in KK-theory: we construct a suitable homotopy at the same time that we construct the isomorphism. One might hope to prove that if A and B are separable C^{∗}-algebras, and if φ_{0} : A → B and ψ_{0} : B → A are homomorphisms such that ψ_{0} ∘ φ_{0} is asymptotically unitarily equivalent to id_{A} and φ_{0} ∘ψ_{0} is asymptotically unitarily equivalent to id_{B}, then there is an isomorphism φ : A → B which is asymptotically unitarily equivalent to φ_{0}, or at least is homotopic to φ_{0}. Unfortunately, we have not been able to prove this; the arguments in the proof below don’t seem to quite give such a result.

4.2.1 Theorem. Let A and B be separable nuclear unital purely infinite simple C^{∗}-algebras, and suppose that there is an invertible element η ∈ KK^{0}(A, B). Then there is an isomorphism φ : K ⊗ A → K ⊗ B such [φ] = η in KK^{0}(A, B).

Proof: Theorems 3.2.8 and 2.1.5 provide a full asymptotic morphism α : A → K ⊗ B whose class in KK^{0}(A, B) is η. By Corollary 2.3.10, we may in fact take α to be a homomorphism. Let μ : K ⊗ K → K be an isomorphism, and set φ_{0} = (μ ⊗ id_{B}) ∘ (id_{K} ⊗ α). Then φ_{0} is a nonzero (hence full) homomorphism from K ⊗ A to K ⊗ B whose class in KK^{0}(A, B) is also η. Similarly, there is a full homomorphism ψ_{0} : K ⊗ B → K ⊗ A whose class in KK^{0}(B, A) is η^{−1}. It follows from Theorems 4.1.3 and 2.1.5 that ψ_{0} ∘ φ_{0} is homotopic to id_{K⊗A} and φ_{0} ∘ ψ_{0} is homotopic to id_{K⊗B}.

We now construct homomorphisms φ^{(n)} : K ⊗A → K ⊗B, ψ^{(n)} : K ⊗B → K ⊗ A, homotopies α ↦ φ_{α}^{(n)} (for α ∈ [0, 1]) of homomorphisms from K ⊗ A to K ⊗B, and finite subsets F_{n} ⊂ K ⊗A and G_{n} ⊂ K ⊗B such that the following

Nuclear Purely Infinite Simple C^{∗}-Algebras 101

conditions are satisfied:

(1) φ^{(0)} = φ_{0}.

(2) Each φ^{(n)} is of the form a ↦ vφ_{0}(a)v^{∗} for some suitable v ∈ U_{0}((K ⊗ B)^{+}), and similarly each ψ^{(n)} is of the form b ↦ uφ_{0}(b)u^{∗} for some suitable u ∈ U_{0}((K ⊗ A)^{+}).

(3) F_{0} ⊂ F_{1} ⊂ ⋯ and ⋃_{n=0}^{∞} F_{n} is dense in K ⊗ A, and similarly G_{0} ⊂ G_{1} ⊂ ⋯ and ⋃_{n=0}^{∞} G_{n} is dense in K ⊗ B.

(4) φ^{(n)}(F_{n}) ⊂ G_{n} and ψ^{(n)}(G_{n}) ⊂ F_{n+1}.

(5) ∥ψ^{(n)} ∘ φ^{(n)}(a) − a∥ < 2^{−n} for a ∈ F_{n} and ∥φ^{(n+1)} ∘ ψ^{(n)}(b) − b∥ < 2^{−n} for b ∈ G_{n}.

(6) ∥φ_{α}^{(n+1)}(a) − φ_{α}^{(n)}(a)∥ < 2^{−n} for a ∈ F_{n} and α ∈ [0, 1].

(7) φ_{α}^{(n)} = φ_{0} for α ≥ 1 − 2^{−n} and φ_{0}^{(n)} = φ^{(n)}. This will yield the following approximately commutative diagram:

[figure: A; A; A; A; B; B; B; B; ⋯; ⋯; id_{A}; id_{A}; id_{A}; id_{A}; id_{A}; id_{B}; id_{B}; id_{B}; id_{B}; id_{B}; φ^{(0)}; φ^{(1)}; φ^{(n−1)}; φ^{(n)}; ψ^{(0)}; ψ^{(1)}; ψ^{(n−2)}; ψ^{(n−1)}; ∩; ∪; ∩; ∪; ∩; ∪; ∩; ∪; F_{0}; G_{0}; F_{1}; G_{1}; F_{n−1}; G_{n−1}; F_{n}; G_{n}]

The diagram will remain approximately commutative if we replace each φ^{(n)} by φ_{α}^{(n)} (with α ∈ [0, 1] fixed) and delete the diagonal arrows.

The proof is by induction on n. We start by choosing finite sets F_{0}^{(0)} ⊂ F_{1}^{(0)} ⊂ ⋯ ⊂ K ⊗ A and G_{0}^{(0)} ⊂ G_{1}^{(0)} ⊂ ⋯ ⊂ K ⊗ B

such that \overline{⋃_{n=0}^{∞} F_{n}^{(0)}} = K ⊗ A and \overline{⋃_{n=0}^{∞} G_{n}^{(0)}} = K ⊗ B. For the initial step of the induction, we take F_{0} = F_{0}^{(0)}, φ^{(0)} = φ_{α}^{(0)} = φ_{0}, and G_{0} = G_{0}^{(0)} ∪ φ^{(0)}(F_{0}). We then assume we are given F_{k}, φ^{(k)}, G_{k}, and φ_{α}^{(k)} for 0 ≤ k ≤ n and ψ^{(k)} for 0 ≤ k ≤ n − 1, and we construct ψ^{(n)}, F_{n+1}, φ^{(n+1)}, G_{n+1}, and α ↦ φ_{α}^{(n+1)}. That is, we are given the diagram above through the column containing F_{n} and G_{n}, as well as the corresponding homotopies φ^{(k)}, and we construct the

Documenta Mathematica 5 (2000) 49–114

next rectangle (consisting of two triangles) and the corresponding homotopy φ^{(n+1)}.

Define σ : K ⊗ A → C([0, 1]) ⊗ K ⊗ A by σ(a)(α) = ψ_{0}(φ_{α}^{(n)}(a)). Note that σ is homotopic to a ↦ 1⊗ψ_{0}(φ_{0}(a)), and so has the same class in KK-theory as a ↦ 1 ⊗ a. Lemma 4.1.2 provides a unitary path (α, t) ↦ u_{α,t} ∈ U_{0}((K ⊗ A)^{+}) such that

lim_{t→∞} sup_{α∈[0,1]} ∥u_{α,t}ψ_{0}(φ_{α}^{(n)}(a))u_{α,t}^{∗} − a∥ = 0

for all a ∈ K ⊗ A. Next, define an asymptotic morphism τ from K ⊗ B to C([0, 1]) ⊗ K ⊗ B by τ_{t}(b)(α) = φ_{0}(u_{α,t}ψ_{0}(b)u_{α,t}^{∗}). Then τ is homotopic to b ↦ 1⊗φ_{0}(ψ_{0}(b)), and so has the same class in KK-theory as b ↦ 1⊗b. Again by Lemma 4.1.2, there is a unitary path (α, t) ↦ v_{α,t} ∈ U_{0}((K ⊗ B)^{+}) such that

lim_{t→∞} sup_{α∈[0,1]} ∥v_{α,t}φ_{0}(u_{α,t}ψ_{0}(b)u_{α,t}^{∗})v_{α,t}^{∗} − b∥ = 0

for all b ∈ K ⊗ B.

Since G̃ = G_{n} ∪ ⋃_{α∈[0,1]} φ_{α}^{(n)}(F_{n}) is a compact subset of K ⊗ B, we can choose T so large that

∥v_{α,t}φ_{0}(u_{α,t}ψ_{0}(b)u_{α,t}^{∗})v_{α,t}^{∗} − b∥ < 2^{−(n+1)}

for all b ∈ G̃ and t ≥ T. Increasing T if necessary, we can also require

∥u_{α,t}ψ_{0}(φ_{α}^{(n)}(a))u_{α,t}^{∗} − a∥ < 2^{−(n+1)}

for all a ∈ F_{n} and t ≥ T. Now define

ψ^{(n)}(b) = u_{0,T} ψ_{0}(b)u_{0,T}^{∗} and φ^{(n+1)}(a) = v_{0,T} φ_{0}(a)v_{0,T}^{∗} ,

and

F_{n+1} = F_{n+1}^{(0)} ∪ F_{n} ∪ ψ^{(n)}(G_{n}) and G_{n+1} = G_{n+1}^{(0)} ∪ G_{n} ∪ φ^{(n+1)}(F_{n+1}).

The relevant parts of conditions (2)–(4) are then certainly satisfied. For (5), we have in fact

∥ψ^{(n)} ∘ φ^{(n)}(a) − a∥ = ∥u_{0,T} ψ_{0}(φ_{0}^{(n)}(a))u_{0,T}^{∗} − a∥ < 2^{−(n+1)}

for a ∈ F_{n} by the choice of T, and similarly

∥φ^{(n+1)} ∘ ψ^{(n)}(b) − b∥ = ∥v_{0,T} φ_{0}(u_{0,T} ψ_{0}(b)u_{0,T}^{∗})v_{0,T}^{∗} − b∥ < 2^{−(n+1)}

for b ∈ G_{n}.

Now choose a continuous function f : [0, 1 − 2^{−(n+1)}) → [T, ∞) such that f(α) = T for 0 ≤ α ≤ 1 − 2^{−n} and f(α) → ∞ as α → 1 − 2^{−(n+1)}. Define α ↦ φ_{α}^{(n+1)} by

φ_{α}^{(n+1)}(a) = \{ \begin{matrix}v_{α,f(α)}φ_{0}(a)v_{α,f(α)}^{∗} & 0 ≤ α < 1 − 2^{−(n+1)} \\ φ_{0}(a) & 1 − 2^{−(n+1)} ≤ α ≤ 1. \end{matrix}

We first have to show that the functions α ↦ φ_{α}^{(n+1)}(a) are continuous at 1−2^{−(n+1)} for a ∈ K ⊗A. Set α_{0} = 1−2^{−(n+1)}, and consider α with 1−2^{−n} ≤ α < 1 − 2^{−(n+1)}. By the induction hypothesis, we then have φ_{α}^{(n)}(a) = φ_{0}(a). For a ∈ K ⊗ A, set b = φ_{0}(a); then

∥φ_{α}^{(n+1)}(a) − φ_{α_{0}}^{(n+1)}(a)∥
≤ ∥a − u_{α,f(α)}ψ_{0}(φ_{α}^{(n)}(a))u_{α,f(α)}^{∗}∥
+ ∥v_{α,f(α)}φ_{0}(u_{α,f(α)}ψ_{0}(b)u_{α,f(α)}^{∗})v_{α,f(α)}^{∗} − b∥.

The requirement that f(α) → ∞ as α → 1 − 2^{−(n+1)}, together with the condition of uniformity in α in the limits used in the choices of u_{α,t} and v_{α,t}, implies that both terms on the right converge to 0. So the required continuity holds.

The relevant part of condition (7) is satisfied by definition, so it remains only to check (6). We may assume α < 1 − 2^{−(n+1)}. So let a ∈ F_{n}. Then b = φ_{α}^{(n)}(a) ∈ G̃. So

∥φ_{α}^{(n+1)}(a) − φ_{α}^{(n)}(a)∥
≤ sup_{α∈[0,1],t≥T} ∥v_{α,t}φ_{0}(a)v_{α,t}^{∗} − φ_{α}^{(n)}(a)∥
≤ sup_{α∈[0,1],t≥T} ∥a − u_{α,t}ψ_{0}(φ_{α}^{(n)}(a))u_{α,t}^{∗}∥

+ sup_{α∈[0,1],t≥T} ∥v_{α,t}φ_{0}(u_{α,t}ψ_{0}(b)u_{α,t}^{∗})v_{α,t}^{∗} − b∥

< 2^{−(n+1)} + 2^{−(n+1)} = 2^{−n}.

This proves (6), and finishes the inductive construction. Note that the set ⋃_{n=0}^{∞} F_{n} is dense in K ⊗A because it contains the dense subset ⋃_{n=0}^{∞} F_{n}^{(0)}, and similarly ⋃_{n=0}^{∞} G_{n} is dense in K ⊗ B.

We now define φ : K ⊗ A → K ⊗ B by φ(a) = lim_{n→∞} φ^{(n)}(a), and define ψ : K ⊗ B → K ⊗ A and the homotopy φ̃ : K ⊗ A → C([0, 1]) ⊗ K ⊗ B analogously. As in Section 2 of [20], these limits all exist and define homomorphisms; moreover, ψ ∘ φ = id_{K⊗A}, φ ∘ ψ = id_{K⊗B}, φ̃_{0} = φ, and φ̃_{1} = φ_{0}. So φ is an isomorphism from K ⊗ A to K ⊗ B which is homotopic to φ_{0} and therefore satisfies [φ] = η in KK^{0}(A, B).

4.2.2 Corollary. Let A and B be separable nuclear unital purely infinite simple C^{∗}-algebras, and suppose that there is an invertible element η ∈ KK^{0}(A, B) such that [1_{A}]×η = [1_{B}]. Then there is an isomorphism φ : A → B such [φ] = η in KK^{0}(A, B).

Proof: The previous theorem provides an isomorphism α : K ⊗ A → K ⊗ B such that [α] = η in KK^{0}(A, B). Choose a rank one projection e ∈ K. Then [α(e ⊗ 1_{A})] = [1_{A}] × η = [e ⊗ 1_{B}] in K_{0}(B). Since K ⊗ B is purely infinite simple, it follows that there is a unitary u ∈ (K ⊗ B)^{+} such that uα(e ⊗ 1_{A})u^{∗} = e ⊗ 1_{B}. Define φ(a) = uα(e ⊗ a)u^{∗}, regarded as an element of (e ⊗ 1_{B})(K ⊗ B)(e ⊗ 1_{B}) = B.

The remaining corollaries require some hypotheses on the Universal Coefficient Theorem. (See [53].) The following terminology is convenient.

4.2.3 Definition. Let A and D be separable nuclear C^{∗}-algebras. We say that the pair (A, D) satisfies the Universal Coefficient Theorem if the sequence

0 ⟶ Ext_{1}^{Z}(K_{∗}(A), K_{∗}(D)) ⟶ KK^{0}(A, D) ⟶ Hom(K_{∗}(A), K_{∗}(D)) ⟶ 0

of Theorem 1.17 of [53] is defined and exact. (Note that the second map is always defined, and the first map is the inverse of a map that is always defined.) We further say that A satisfies the Universal Coefficient Theorem if (A, D) does for every separable C^{∗}-algebra D.

4.2.4 Theorem. Let A and B be separable nuclear purely infinite simple C^{∗}- algebras which satisfy the Universal Coefficient Theorem. Assume that A and B are either both unital or both nonunital. If there is a graded isomorphism α : K_{∗}(A) → K_{∗}(B) which (in the unital case) satisfies α_{∗}([1_{A}]) = [1_{B}], then there is an isomorphism φ : A → B such that φ_{∗} = α.

Proof: The proof of Proposition 7.3 of [53] shows that there is a KK-equivalence η ∈ KK^{0}(A, B) which induces α. Now use Theorem 4.2.1 or Corollary 4.2.2 as appropriate.

This theorem gives all the classification results of [48], [49], [34], [22], [35], [51], [36], [44], and [52]. Of course, we have used the main technical theorem of [48], as well as substantial material from [35], in the proof. We do not obtain anything new about the Rokhlin property of [8]; indeed, our results show that the C^{∗}-algebras of [51] are classifiable as long as they are purely infinite and simple, regardless of whether the Rokhlin property is satisfied. On the other hand, the Rokhlin property has been verified in many cases; see [30] and [31].

We finish this section by giving some further corollaries. Let C be the “classifiable class” given in Definition 5.1 of [22], and let N denote the bootstrap category of [53], for which the Universal Coefficient Theorem was shown to hold (Theorem 1.17 of [53]).

4.2.5 Theorem. Let G_{0} and G_{1} be countable abelian groups, and let g ∈ G_{0}. Then:

(1) There is a separable nuclear unital purely infinite simple C^{∗}-algebra algebra A ∈ N such that

(K_{0}(A), [1_{A}], K_{1}(A)) ≅ (G_{0}, g, G_{1}).

(2) There is a separable nuclear nonunital purely infinite simple C^{∗}-algebra A ∈ N such that

(K_{0}(A), K_{1}(A)) ≅ (G_{0}, G_{1}).

Proof: The construction of Theorem 5.6 of [22] gives algebras which are easily seen to be in N .

4.2.6 Corollary. Every C^{∗}-algebra in C is in N . Every purely infinite simple C^{∗}-algebra in N , and more generally every separable nuclear purely infinite simple C^{∗}-algebra satisfying the Universal Coefficient Theorem, is in C.

Proof: The first part follows immediately from the previous theorem, since it follows from the definition of C that any A ∈ C must be isomorphic to the C^{∗}-algebra of that theorem with the same K-theory. The second part follows from Theorem 4.2.4, since Theorem 1.17 of [53] states that every C^{∗}-algebra in N satisfies the Universal Coefficient Theorem.

4.2.7 Corollary. Let A ∈ C, and let B be a separable nuclear unital simple C^{∗}-algebra which satisfies the Universal Coefficient Theorem. (In particular, B could be a unital simple C^{∗}-algebra in N .) Then A ⊗ B ∈ C.

Proof: The C^{∗}-algebra A ⊗ B is separable, nuclear, unital, and simple, and Theorem 7.7 of [53] (and the remark after this theorem) shows that it satisfies the Universal Coefficient Theorem. Furthermore, A is approximately divisible by Corollary 2.1.6, and it follows from the remark after Theorem 1.4 of [6] that A ⊗ B is approximately divisible. Clearly A ⊗ B is infinite, so it is purely infinite by Theorem 1.4 (a) of [6]. The result now follows from the previous corollary.

4.2.8 Corollary. The class C is closed under tensor products.

4.2.9 Corollary. For any m, n ≥ 2, we have O_{m} ⊗ O_{n} ∈ C. In particular, if m − 1 and n − 1 are relatively prime, then O_{m} ⊗ O_{n} ≅ O_{2}.

4.2.10 Corollary. Let A_{1} and A_{2} be two higher dimensional noncommutative toruses of the same dimension, and let B be any simple Cuntz-Krieger algebra. Then A_{1} ⊗ B ≅ A_{2} ⊗ B.

Proof: The Künneth formula [55] shows that A_{1} ⊗B and A_{2} ⊗B have the same K-theory.

4.2.11 Theorem. Let A be a separable nuclear unital purely infinite simple C^{∗}-algebra satisfying the Universal Coefficient Theorem. Let A^{op} be the opposite algebra, that is, A with the multiplication reversed but all other operations the same. Then A ≅ A^{op}.

Proof: The identity map from A to A^{op} is an antiisomorphism which induces an isomorphism on K-theory sending [1_{A}] to [1_{A^{op}}]. Also, the pair (A^{op}, B) (for any separable B) always satisfies the Universal Coefficient Theorem, because (A, B^{op}) does.

By way of contrast, we note that Connes has shown [10] that there is a type III factor not isomorphic to its opposite algebra. It is also known (although apparently not published) that there are nonsimple separable nuclear (even type I) C^{∗}-algebras not isomorphic to their opposite algebras.

4.3 Nonclassification

In this subsection, we give some results which show how badly the classification theorem fails if the algebras are not nuclear. The results are mostly either proved elsewhere or follow fairly easily from results proved by other people. There are three main results. First, nonnuclear separable purely infinite simple C^{∗}-algebras need not be approximately divisible in the sense of [6], but whenever A is a purely infinite simple C^{∗}-algebra, then O_{∞} ⊗ A is an approximately divisible purely infinite simple C^{∗}-algebra with exactly the same K-theoretic invariants. Second, there are infinitely many mutually nonisomorphic approximately divisible separable exact unital purely infinite simple C^{∗}-algebras A satisfying K_{∗}(A) = 0. Finally, given arbitrary countable abelian groups G_{0} and G_{1}, and g ∈ G_{0}, there are uncountably many mutually nonisomorphic approximately divisible separable unital purely infinite simple C^{∗}-algebras A satisfying K_{j}(A) ≅ G_{j} with [1] ↦ g_{0}. Unfortunately these algebras are not exact, and it remains unknown whether the same is true with the additional requirement of exactness.

The first result is taken straight from a paper of Dykema and Rørdam.

4.3.1 Theorem. ([18], Theorem 1.4) There exists a separable unital purely infinite simple C^{∗}-algebra which is not approximately divisible.

4.3.2 Remark. In fact, there exists a separable unital purely infinite simple C^{∗}-algebra A which is not approximately divisible and such that K_{∗}(A) = 0.

One way to see this is to modify the proof of Proposition 1.3 of [18] so as to ensure that K_{∗}(A_{n}) → K_{∗}(B) is injective for all n. This is done by enlarging the set X_{n+1} in the proof so as to include appropriate partial isometries (implementing equivalences between projections) and paths of unitaries (implementing triviality of classes of unitaries in K_{1}). See the proof of Theorem 4.3.11 below for this argument in a related context.

The second result is a fairly easy consequence of a computation of Cowling and Haagerup and of unpublished work of Haagerup. The key invariant is described in the following definition. I am grateful to Uffe Haagerup for explaining the properties of this invariant and where to find proofs of them.

4.3.3 Definition. (Haagerup [23]; also see Section 6 of [12].) Let A be a C^{∗}-algebra. Define Λ(A) to be the infimum of numbers C such that there is a net of finite rank operators T_{α} : A → A for which ∥T_{α}(a) − a∥ → 0 for all a ∈ A and the completely bounded norms satisfy sup_{α} ∥T_{α}∥_{cb} ≤ C. Note that Λ(A) = ∞ if no such C exists, that is, if A does not have the completely bounded approximation property.

There is a similar definition for von Neumann algebras, in which T_{α}(a) is required to converge to a in the weak operator topology. (See [23] and Section 6 of [12].) There is also a definition of Λ(G) for a locally compact group G, using completely bounded norms of multipliers of G which converge to 1 uniformly on compact sets; see [23] and Section 1 of [12]. We do not formally state the definitions, but we recall the following theorems from [23] (restated as Propositions 6.1 and 6.2 of [12]):

4.3.4 Theorem. Let Γ be a discrete group, and let C_{r}^{∗}(Γ) and W^{∗}(Γ) be its reduced C^{∗}-algebra and von Neumann algebra respectively. Then Λ(Γ) = Λ(C_{r}^{∗}(Γ)) = Λ(W^{∗}(Γ)).

4.3.5 Theorem. Let G be a second countable locally compact group, and let Γ be a lattice in G. Then Λ(Γ) = Λ(G).

In Section 6 of [12], Cowling and Haagerup exhibit type II_{1} factors M_{n} with Λ(M_{n}) = 2n − 1. Using the same results on groups, we exhibit simple C^{∗}-algebras with the same values of Λ.

4.3.6 Proposition. Let Γ_{n}^{0} be as in Corollary 6.6 of [12]. Then A_{n} = C_{r}^{∗}(Γ_{n}^{0}) is a simple separable unital C^{∗}-algebra which satisfies Λ(A_{n}) = 2n − 1.

We recall that Γ_{n}^{0} is the quotient by its center of a particular lattice Γ_{n} in the simple Lie group Sp(n, 1).

Proof of Proposition 4.3.6: It is shown in the proof of Corollary 6.6 of [12] that Λ(Γ_{n}^{0}) = 2n−1. (This follows from the computation Λ(Sp(n, 1)) = 2n−1, which is the main result of [12], together with Theorem 4.3.5 above and Proposition 1.3 (c) of [12].) Therefore Λ(A_{n}) = 2n − 1 by Theorem 4.3.4. Clearly A_{n} is separable and unital. Simplicity of A_{n} follows from Theorem 1 of [2], applied to the quotient of Sp(n, 1) by its center, because (as observed in the introduction to [2]) lattices satisfy the density hypothesis of that theorem.

The algebras A_{n} are not purely infinite, and their K-theory seems to be unknown. So we will tensor them with O_{2}. For this, we need the following result.

4.3.7. Lemma. Let A be any C^{∗}-algebra, and let B be unital and nuclear. Then Λ(A ⊗ B) = Λ(A).

For von Neumann algebras, it is known [57] that Λ(M ⊗N) = Λ(M)Λ(N). We presume, especially in view of Remark 3.5 of [57], that the analogous statement is true for C^{∗}-algebras as well. However, the special case in the lemma is sufficient here.

Proof of Lemma 4.3.7: If S : A_{1} → A_{2} and T : B_{1} → B_{2} are completely bounded, then the map S ⊗_{min} T : A_{1} ⊗_{min} B_{1} → A_{2} ⊗_{min} B_{2} is completely bounded, and satisfies ∥S ⊗_{min} T ∥_{cb} = ∥S∥_{cb}∥T ∥_{cb} by Theorem 10.3 of [40]. In Definition 4.3.3, one need only consider elements a of a dense subset, and so it follows that Λ(A ⊗_{min} B) ≤ Λ(A)Λ(B) for any C^{∗}-algebras A and B. For B nuclear, we have Λ(B) = 1, so Λ(A ⊗ B) ≤ Λ(A).

For the reverse inequality, let R_{α} : A⊗B → A⊗B be finite rank operators such that ∥R_{α}(x) − x∥ → 0 for all x ∈ A ⊗ B. Choose any state ω on B, and define T_{α} : A → A by T_{α}(a) = (id_{B} ⊗ ω) ∘ R_{α}(a ⊗ 1). Theorem 10.3 of [40] implies that ∥T_{α}∥_{cb} ≤ ∥R_{α}∥_{cb}. Also, clearly ∥T_{α}(a) − a∥ → 0 for all a ∈ A. So Λ(A) ≤ Λ(A ⊗ B).

4.3.8 Theorem. There exist infinitely many mutually nonisomorphic separable exact unital purely infinite simple C^{∗}-algebras B satisfying K_{∗}(B) = 0 and O_{∞} ⊗ B ≅ B. In particular, these algebras are approximately divisible in the sense of [6].

Proof: Let A_{n} = C_{r}^{∗}(Γ_{n}^{0}) as in Proposition 4.3.6. Set B_{n} = O_{2} ⊗ A_{n}. Clearly B_{n} is separable and unital. Furthermore, B_{n} is purely infinite simple by the proof of Corollary 4.2.7. We have O_{∞} ⊗ B_{n} ≅ B_{n} because O_{∞} ⊗ O_{2} ≅ O_{2}. The algebras B_{n} are mutually nonisomorphic because Λ(B_{n}) = 2n − 1, by the previous lemma and Proposition 4.3.6.

It remains to check exactness. The proof of Corollary 3.12 of [17] shows that if Λ(A) is finite, then A has the slice map property (as defined, for example, in Remark 9 of [59], where it is called Property S), and this property implies exactness (see, for example, Section 2.5 of [60]).

Our third result is based on the theorem of Junge and Pisier that for n ≥ 3 the collection of n-dimensional operator spaces is not separable in the completely bounded analog of the Banach-Mazur distance.

4.3.9 Definition. ([25]) Let E and F be operator spaces of the same finite dimension. Then

d_{cb}(E, F) = inf{∥T ∥_{cb}∥T^{−1}∥_{cb} : T is a linear bijection from E to F },

and δ_{cb}(E, F) = log(d_{cb}(E, F)).

4.3.10 Theorem. (Theorem 2.3 of [25]) Let OS_{n} be the set of all complete isometry classes of n-dimensional operator spaces. Let n ≥ 3. Then (OS_{n}, δ_{cb}) is an inseparable metric space.

4.3.11 Theorem. Let G_{0} and G_{1} be countable abelian groups, and let g ∈ G_{0}. Then there exist uncountable many mutually nonisomorphic separable unital purely infinite simple C^{∗}-algebras A, each with K_{0}(A) ≅ G_{0} in such a way that [1] ↦ g and K_{1}(A) ≅ G_{1}, and each satisfying O_{∞} ⊗ A ≅ A.

Proof: If A is a separable C^{∗}-algebra, then the set of (complete isometry classes of) n-dimensional operator subspaces of A is separable (by Proposition 2.6 (a) of [25]). By the previous theorem, it therefore suffices to show that if E is a finite dimensional operator space then there exists a C^{∗}-algebra B having the properties claimed in the theorem and such that E is completely isometric to a subspace of B.

Since E is a finite dimensional operator space, it is a subspace of a separable C^{∗}-algebra A. Represent A on a separable Hilbert space H with infinite multiplicity, and follow this representation with the quotient map from L(H) to the Calkin algebra Q. This gives a completely isometric embedding of E in Q. For convenience, we identify E with its image. Let u ∈ Q be the image of the unilateral shift; note that [u] generates K_{1}(Q) and that K_{0}(Q) = 0. Let B_{0} = C^{∗}(E, 1, u) ⊂ Q. We now construct by induction an increasing sequence B_{0} ⊂ B_{1} ⊂ B_{2} ⊂ ⋯ ⊂ Q of separable C^{∗}-algebras such that B_{2n+1} is simple and such that every nonzero projection in B_{2n−1} is Murray-von Neumann equivalent to 1 in B_{2n}, every selfadjoint element of B_{2n−1} is a limit of selfadjoint elements of B_{2n} with finite spectrum, and every unitary in U(B_{2n−1}) ∩ U_{0}(Q) is homotopic to 1 in B_{2n}.

Given B_{2n}, we choose B_{2n+1} to be any separable simple C^{∗}-algebra with B_{2n} ⊂ B_{2n+1} ⊂ Q. Such a subalgebra exists by Proposition 2.2 of [3] and the simplicity of Q. Given B_{2n−1}, we note that it suffices to have the required elements of B_{2n} only for countable dense subsets S_{1} of the nonzero projections in B_{2n−1}, S_{2} of the selfadjoint elements in B_{2n−1}, and S_{3} of the unitaries in U(B_{2n−1})∩U_{0}(Q). For each p ∈ S_{1}, since p is Murray-von Neumann equivalent to 1 in Q, we can choose an isometry v ∈ Q such that v^{∗}v = 1 and vv^{∗} = p. Let T_{1} be the set of all these as p runs through S_{1}. For each a ∈ S_{2}, since Q has real rank zero, there is a sequence (b_{n}) in Q consisting of selfadjoint elements with finite spectrum such that b_{n} → a. Let T_{2} be the set of all terms of all such sequences as a runs through S_{2}. For each u ∈ S_{3}, since u ∈ U_{0}(Q), there is a unitary path t ↦ v(t) in Q with v(0) = 1 and v(1) = u. Let T_{3} consist of all v(t) for t ∈ [0, 1] ∩ Q as u runs through S_{3}. Then take B_{2n} to be the C*-subalgebra of Q generated by B_{2n−1} and T_{1} ∪ T_{2} ∪ T_{3}. This subalgebra is separable because B_{2n−1} is separable and T_{1} ∪ T_{2} ∪ T_{3} is countable.

Now set B = \overline{⋃_{n=0}^{∞} B_{n}}. Then B is simple because it is the direct limit of the simple C^{∗}-algebras B_{2n+1}. From the construction of B_{2n}, it is clear that B is unital and separable, contains the operator space E, has real rank zero, that all nonzero projections in B are Murray-von Neumann equivalent to 1, and that U(B) ∩ U_{0}(Q) ⊂ U_{0}(B). The third and fourth properties imply that B is purely infinite and K_{0}(B) = 0. The last property implies that K_{1}(B) → K_{1}(Q) is injective. But this map is also surjective, since B_{0} contains a unitary whose class generates K_{1}(Q). So K_{1}(B) ≅ Z.

Taking A = O_{∞} ⊗ B (which has the same K-theory by the Künneth formula [55]), we obtain the statement of the theorem for the special case G_{0} = 0, g = 0, and G_{1} = Z. For the general case, choose (by Theorem 4.2.5) a separable nuclear unital purely infinite simple C^{∗}-algebra D satisfying the Universal Coefficient Theorem and such that K_{0}(D) ≅ G_{1} and K_{1}(D) ≅ G_{0}. (We don’t actually need D to be purely infinite here, but it must be in the bootstrap category of [55].) Then D ⊗ B is purely infinite and simple, and has the right K-theory by the Künneth formula, except that [1] = 0. Choose a projection p ∈ D ⊗ B such that the isomorphism K_{0}(D ⊗ B) ≅ G_{0} sends [p] to g. Then the C^{∗}-algebra A = O_{∞} ⊗ p(D ⊗ B)p satisfies all the conditions of the theorem and contains the given operator space E.

4.3.12 Remark. Simplicity and pure infiniteness of \overline{⋃_{n=0}^{∞} B_{n}} in the proof above can also be arranged by the method of the proof of Proposition 1.3 of [18]. Versions of the construction here have been used many times before.

4.3.13 Remark. The invariant used here, the set of finite dimensional operator spaces contained in A, does not distinguish between any two separable exact purely infinite simple C^{∗}-algebras. (Any separable exact C^{∗}-algebra embeds in O_{2} by Theorem 2.8 of [29], and O_{2} embeds in any purely infinite simple C^{∗}-algebra.) Therefore, for given K-theory, at most one of the C^{∗}-algebras proved above to be nonisomorphic can be exact.

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N. Christopher Phillips Department of Mathematics University of Oregon Eugene OR 97403-1222 USA ncp@darkwing.uoregon.edu