Stability of C^{∗}-Algebras is Not a Stable Property

Mikael Rørdam

Received: November 28, 1997

Communicated by Joachim Cuntz

Abstract. We show that there exists a C^{∗}-algebra B such that M_{2}(B) is stable, but B is not stable. Hence stability of C^{∗}-algebras is not a stable property. More generally, we find for each integer n ≥ 2 a C^{∗}-algebra B so that M_{n}(B) is stable and M_{k}(B) is not stable when 1 ≤ k < n. The C^{∗}-algebras we exhibit have the additional properties that they are simple, nuclear and of stable rank one.

The construction is similar to Jesper Villadsen’s construction in [7] of a simple C^{∗}-algebra with perforation in its ordered K_{0}-group.

1991 Mathematics Subject Classification: 46L05, 46L35, 19K14

Keywords: Stable C^{∗}-algebras, perforation in K_{0}, scaled ordered Abelian groups.

1 Introduction

A C^{∗}-algebra A is said to be stable if A ≅ A ⊗ K, where K is the C^{∗}-algebra of compact operators on a separable, infinite dimensional Hilbert space. The problem of deciding which C^{∗}-algebras are stable relates to structure problems of simple C^{∗}- algebras. For example, as shown in [3, Proposition 5.2], if all non-unital hereditary sub-C^{∗}-algebras of a given C^{∗}-algebra A are stable, and if A is simple and not of type I, then A must be purely infinite. It was also remarked in [3, Proposition 5.1] that an AF-algebra is stable if and only if it admits no bounded (densely defined) traces, and it was asked if a similar characterization might hold in general. In more detail, is a C^{∗}-algebra A stable if and only if A admits no bounded (quasi-)trace and no quotient of A is unital?

It is a consequence of the examples produced in this article that the answer to this question is no. Indeed, let A be a C^{∗}-algebra such that M_{2}(A) is stable and A is not stable. Then M_{2}(A) admits no bounded (quasi-)trace, and no quotient of M_{2}(A) is unital. This is easily seen to imply that A admits no bounded (quasi-)trace, and that no quotient of A is unital.

Jesper Villadsen gave in [7] the first examples of simple C^{∗}-algebras whose ordered K_{0}-groups have perforation. As shown in Proposition 3.3, the examples constructed here must also have perforation in their K_{0}-group (at least when they admit an approximate unit consisting of projections). We shall in this article make extensive use of the techniques developed by Villadsen.

2 A preliminary result

Let A be a C^{∗}-algebra and consider the set Γ(A) consisting of those integers n ≥ 1 where M_{n}(A) is stable. The result below shows that this set must be either empty, ℕ, or equal to {n, n + 1, n + 2, … } for some n ≥ 2. Clearly, the empty set and ℕ arise as Γ(A) for appropriate C^{∗}-algebras A. The main result of this article (Theorem 5.3) shows that the remaining sets are also realized.

Proposition 2.1 Let A be a σ-unital C^{∗}-algebra, let n ≥ 1 be an integer, and suppose that M_{n}(A) is stable. Then M_{n+1}(A) is stable.

Proof: By [3, Theorem 2.1 and Proposition 2.2] it suffices to show that one for all positive elements a ∈ M_{n+1}(A) and all ε > 0 can find positive elements b, c ∈ M_{n+1}(A) with ∥a−b∥ ≤ ε, ∥bc∥ ≤ ε, and b ∼ c (i.e. b = x^{∗}x and c = xx^{∗} for some x ∈ M_{n+1}(A)). To obtain this it suffices to find positive elements e, f ∈ M_{n+1}(A)^{+} with e ∼ f, e ⊥ f, and ea close to a. Indeed, if e = x^{∗}x and f = xx^{∗}, then set y = xa^{1/2}, and note that y^{∗}y is close to a and that (yy^{∗})(y^{∗}y) is small.

Now,

a = ( \begin{matrix}a_{1} & z \\ z^{∗} & a_{2} \end{matrix} ) ,

where a_{1} ∈ M_{n}(A)^{+}, a_{2} ∈ A^{+} and z ∈ M_{n,1}(A). Let ε > 0, and let φ_{ε}: ℝ^{+} → [0, 1] be a continuous function which is zero on [0, ε/2] and equal to 1 on [ε, ∞). Set

e′ = ( \begin{matrix}φ_{ε}(a_{1}) & 0 \\ 0 & φ_{ε}(a_{2}) \end{matrix} ) .

Then e′a is close to to a if ε > 0 is small.

Since M_{n}(A) is stable, we can find positive elements e_{1}, f_{1}, f_{2} ∈ M_{n}(A) and e_{2} ∈ A such that e_{1} ∼ f_{1}, e_{2} ∼ f_{2} (in the sense that e_{2} = x^{∗}x and f_{2} = xx^{∗} for some x ∈ M_{n,1}(A)), e_{1}, f_{1}, f_{2} are mutually orthogonal, e_{1} is close to φ_{ε}(a_{1}), and e_{2} is close to φ_{ε}(a_{2}). Set

e = ( \begin{matrix}e_{1} & 0 \\ 0 & e_{2} \end{matrix} ) , f = ( \begin{matrix}f_{1} + f_{2} & 0 \\ 0 & 0 \end{matrix} ) .

Then ea is close to a, e ∼ f, and e ⊥ f as desired. □

3 Stability and the scale of K_{0}

We investigate in this section the connection between the scaled ordered group of a C^{∗}-algebra and stability of matrix algebras over the C^{∗}-algebra. Recall that if A is a C^{∗}-algebra, then

Stability of C^{∗}-Algebras is Not a Stable Property 377

K_{0}(A)^{+} = {[p]_{0} | p ∈ P (A ⊗ K)} ⊆ K_{0}(A), Σ(A) = {[p]_{0} | p ∈ P (A)} ⊆ K_{0}(A)^{+}, where P (A ⊗ K) and P (A) denote the set of projections in A ⊗ K, respectively, A.

One can in some cases see from the triple (K_{0}(A), K_{0}(A)^{+}, Σ(A)) if A is stable. A C^{∗}-algebra A is said to have the cancellation property if p + r ∼ q + r implies that p ∼ q for all projections p, q, r ∈ A⊗K with p ⊥ r and q ⊥ r. If A has the cancellation property, then [p]_{0} = [q]_{0} in K_{0}(A) implies p ∼ q for all projections p, q ∈ A ⊗ K. Recall also that A has the cancellation property if A is of stable rank one (see [1, Proposition 6.5.1]).

Proposition 3.1 Let A be a C^{∗}-algebra with the cancellation property and with a countable approximate unit consisting of projections. Then A is stable if and only if Σ(A) = K_{0}(A)^{+}.

Proof: The “only if” part is trivial. To show the “if” part, assume that Σ(A) = K_{0}(A)^{+}. By [3, Theorem 3.3] it suffices to show that for each projection p ∈ A there exists a projection q ∈ A with p ∼ q and p ⊥ q. Let a projection p ∈ A be given. By the assumptions that A has an approximate unit consisting of projections, and Σ(A) = K_{0}(A)^{+}, there exist projections e, f ∈ A such that [e]_{0} = 2[p]_{0} = [p ⊕ p]_{0}, e ≤ f and p ≤ f. Since A has the cancellation property, this implies that e ∼ p ⊕ p, which again implies that e = e_{1} + e_{2}, where e_{1} ∼ e_{2} ∼ p. Now, [f − p]_{0} = [f − e_{1}]_{0}, and so p ∼ e_{2} ≤ f − e_{1} ∼ f − p. Hence p is equivalent to a subprojection q of f − p as desired. □

Definition 3.2 A triple (G, G^{+}, Σ) will be called a scaled, ordered abelian group if (G, G^{+}) is an ordered abelian group, and Σ is an upper directed, hereditary, full subset of G^{+}, ie.,

(i) ∀x_{1}, x_{2} ∈ Σ ∃x ∈ Σ : x_{1} ≤ x, x_{2} ≤ x,

(ii) ∀x ∈ G^{+} ∀y ∈ Σ : x ≤ y ⟹ x ∈ Σ,

(iii) ∀x ∈ G^{+} ∃y ∈ Σ ∃k ∈ ℕ : x ≤ ky.

Let (G, G^{+}) be an ordered abelian group, and let Σ_{1} and Σ_{2} be upper directed, hereditary, full subsets of G^{+}. Define Σ_{1} ˆ+ Σ_{2} to be the set of all elements x ∈ G^{+} for which there exist x_{1} ∈ Σ_{1} and x_{2} ∈ Σ_{2} with x ≤ x_{1} + x_{2}. Observe that Σ_{1} ˆ+ Σ_{2} is an upper directed, hereditary, full subset of G^{+}. Denote the k-fold sum Σ ˆ+ Σ ˆ+ ⋯ ˆ+ Σ by kˆ·Σ. Using that Σ is upper directed we see that y ∈ kˆ·Σ if and only if 0 ≤ y ≤ kx for some x ∈ Σ.

If A is a stably finite C^{∗}-algebra with the cancellation property and with an approximate unit consisting of projections, then (K_{0}(A), K_{0}(A)^{+}, Σ(A)) is a scaled, ordered abelian group. If A has these properties, then

(K_{0}(M_{k}(A)), K_{0}(M_{k}(A))^{+}, Σ(M_{k}(A))) ≅ (K_{0}(A), K_{0}(A)^{+}, kˆ·Σ(A)).  (1)

Suppose that n ≥ 2 and that (G, G^{+}, Σ) is a scaled, ordered Abelian group such that (n − 1)ˆ·Σ ≠ G^{+} and nˆ·Σ = G^{+}, and suppose that A is a C^{∗}- algebra of stable rank one and with an approximate unit of projections such that (K_{0}(A), K_{0}(A)^{+}, Σ(A)) ≅ (G, G^{+}, Σ). Then it follows from Proposition 3.1 and (1) that M_{n}(A) is stable and M_{k}(A) is not stable for 1 ≤ k < n.

Recall that an ordered Abelian group (G, G^{+}) is called weakly unperforated if ng ∈ G^{+} \ {0} for some n ∈ ℕ and some g ∈ G implies g ∈ G^{+}.

Proposition 3.3 Let (G, G^{+}, Σ) be a weakly unperforated, scaled, ordered, Abelian group, and suppose that nˆ·Σ = G^{+} for some n ∈ ℕ. Then Σ = G^{+}.

Proof: Let g be an element of G^{+}, and choose a non-zero element u ∈ G^{+}. Since nˆ·Σ = G^{+}, there is an element x ∈ Σ with nx ≥ ng + u. Now, n(x − g) ≥ u > 0, and this entails that x − g ≥ 0, by the assumption that (G, G^{+}) is weakly unperforated. By the hereditary property of Σ we get that g ∈ Σ. Thus Σ = G^{+}. □

We give below an explicit example of a scaled, ordered Abelian group (G, G^{+}, Σ) with Σ ˆ+ Σ = G^{+} and Σ ≠ G^{+}. Note that this ordered group necessarily must be perforated (by Proposition 3.3 above).

It is not known if every (countable) scaled ordered Abelian group is the scaled ordered Abelian group of a C^{∗}-algebra — the problem here lies in realizing the given order structure, not in realizing the given scale. We can therefore not immediately conclude from the example below that there exists a non-stable C^{∗}-algebra B where M_{2}(B) is stable. Actually, it is not known (to the author) if the ordered Abelian group constructed below is the ordered K_{0}-group of any C^{∗}-algebra.

Example 3.4 Let ℤ_{2} denote the group ℤ/2ℤ, and let ℤ_{2}^{(∞)} denote the group of all sequences t = (t_{j})_{j=1}^{∞}, with t_{j} ∈ ℤ_{2} and where t_{j} ≠ 0 only for finitely many j. For each t ∈ ℤ_{2}^{(∞)}, let d(t) be the number of elements in {j ∈ ℕ | t_{j} ≠ 0}. Set

G_{2} = ℤ ⊕ ℤ_{2}^{(∞)},  G_{2}^{+} = {(k, t) | d(t) ≤ k},  Σ_{2} = {(k, t) | d(t) = k}.

Then (G_{2}, G_{2}^{+}, Σ_{2}) is a scaled, ordered Abelian group with Σ_{2} ≠ G_{2}^{+} and Σ_{2} ˆ+ Σ_{2} = G_{2}^{+}. To see this, let e_{j} ∈ ℤ_{2}^{(∞)} be the element which is a generator of ℤ_{2} at the jth coordinate and zero elsewhere, set g_{j} = (1, e_{j}) ∈ G^{+}, and set h_{j} = g_{1} + g_{2} + ⋯ + g_{j}. Then

Σ_{2} = ⋃_{j=1}^{∞}{x ∈ G^{+} | x ≤ h_{j}}.  (2)

The claims made about (G_{2}, G_{2}^{+}, Σ_{2}) are now easy to verify.

Notice that Σ_{2} + Σ_{2} ≠ Σ_{2} ˆ+ Σ_{2}, since for example (3, e_{1} + e_{2}) ∉ Σ_{2} + Σ_{2}. This was pointed out to me by Jacob Hjelmborg, and it shows that the sum of two scales is not a scale in general. □

Example 3.5 Let n ≥ 2 be an integer. Let ℤ_{n}^{(∞)} be the Abelian group of all sequences (t_{j})_{j=1}^{∞} with t_{j} ∈ ℤ_{n} (= ℤ/nℤ), and t_{j} ≠ 0 only for finitely many j. Let e_{j} ∈ ℤ_{n}^{(∞)} be a generator of the jth copy of ℤ_{n}. Then each t ∈ ℤ_{n}^{(∞)} is a sum t = ∑_{j=1}^{∞} r_{j}e_{j} with 0 ≤ r_{j} < n and where r_{j} = 0 for all but finitely many j. Set d(t) = ∑_{j=1}^{∞} r_{j}, and set

G_{n} = ℤ ⊕ ℤ_{n}^{(∞)},  G_{n}^{+} = {(k, t) | d(t) ≤ k},  Σ_{n} = ⋃_{j=1}^{∞}{x ∈ G^{+} | x ≤ h_{j}},

where g_{j} = (1, e_{j}) and h_{j} = g_{1} +g_{2} +⋯+g_{j}. Then (G_{n}, G_{n}^{+}, Σ_{n}) is a scaled, ordered, Abelian group, (n − 1)ˆ·Σ_{n} ≠ G_{n}^{+} and nˆ·Σ_{n} = G_{n}^{+}. □

Adopt the following (standard) notation. If e ∈ M_{n}(A) and f ∈ M_{m}(A) are projections, then let e⊕f denote the projection diag(e, f) ∈ M_{n+m}(A). Write e ∼ f if there is an element v ∈ M_{m,n}(A) with e = v^{∗}v and f = vv^{∗}, and write e ≾ f if e ∼ f_{0} for some subprojection f_{0} of f. Denote the k-fold direct sum e ⊕ e ⊕ ⋯ ⊕ e by e ⊗ 1_{k}. If A has the cancellation property (see the introduction to this section), and if e, f ∈ A are projections, then [e]_{0} ≤ [f]_{0} if and only if e ≾ f.

Proposition 3.6 Let A be a C^{∗}-algebra, let n ≥ 2 be an integer, and suppose that A contains projections e, p_{1}, p_{2}, p_{3}, … that satisfy

(i) e ⊗ 1_{n} ∼ p_{j} ⊗ 1_{n} for all j,

(ii) e is not equivalent to a subprojection of (p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{j}) ⊗ 1_{n−1} for any j.

Set q_{j} = p_{1} ⊕p_{2} ⊕⋯⊕p_{j}, and embed all matrix algebras over A coherently into A⊗K so that q_{j} belongs to A ⊗ K for all j. Set

B = \overline{⋃_{j=1}^{∞} q_{j}(A ⊗ K)q_{j}}.  (3)

Then M_{k}(B) is not stable for 1 ≤ k < n, but M_{n}(B) is stable.

Let H be the subgroup of K_{0}(B) generated by the K_{0}-classes of the projections e, p_{1}, p_{2}, p_{3}, . . .. Assume that B has the cancellation property. Then

(H, H ∩ K_{0}(B)^{+}, H ∩ Σ(B)) ≅ (G_{n}, G_{n}^{+}, Σ_{n}),  (4)

where the triple on the right hand-side is the scaled, ordered, Abelian group defined in Example 3.5.

Proof: Observe that

M_{k}(B) = \overline{⋃_{j=1}^{∞}(q_{j} ⊗ 1_{k})(A ⊗ K)(q_{j} ⊗ 1_{k})},

for each k, and that {q_{j} ⊗ 1_{k}}_{j=1}^{∞} is an approximate unit for M_{k}(B).

To show that M_{k}(B) is not stable for 1 ≤ k < n it suffices by Proposition 2.1 to show that M_{n−1}(B) is not stable.

If M_{n−1}(B) were stable, then there would exist a projection q ∈ M_{n−1}(B) such that q ∼ p_{1} ⊗ 1_{n−1} and q ⊥ p_{1} ⊗ 1_{n−1}. (This is rather easy to see directly, and one can also obtain this from [3, Theorem 3.3].) Since {q_{j} ⊗ 1_{n−1} − p_{1} ⊗ 1_{n−1}}_{j=1}^{∞} is an approximate unit for (1 − p_{1} ⊗ 1_{n−1})M_{n−1}(B)(1 − p_{1} ⊗ 1_{n−1}), there is a j, so that q is equivalent to a subprojection of q_{j} ⊗ 1_{n−1} − p_{1} ⊗ 1_{n−1} (= (p_{2} ⊕ p_{3} ⊕ ⋯ ⊕ p_{j}) ⊗ 1_{n−1}). By assumption (i),

e ≾ e ⊗ 1_{n} ∼ p_{1} ⊗ 1_{n} ≾ (p_{1} ⊗ 1_{n−1}) ⊕ (p_{1} ⊗ 1_{n−1}) ≾ (p_{1} ⊗ 1_{n−1}) ⊕ q
≾ (p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{j}) ⊗ 1_{n−1},

in contradiction with assumption (ii).

We proceed to show that M_{n}(B) is stable. By (i), q_{j} ⊗ 1_{n} is equivalent to the direct sum of e ⊗ 1_{n} with itself j times. It follows quite easily from this that M_{n}(B) is stable. We can also use [3, Theorem 3.3] to obtain this conclusion by showing that there for each projection p in M_{n}(B) exists a projection q in M_{n}(B) with p ∼ q and p ⊥ q. One can here reduce to the case where p is a subprojection of q_{j} ⊗ 1_{n} for some j, and the result then follows from the fact that q_{2j} ⊗ 1_{n} − q_{j} ⊗ 1_{n} ∼ q_{j} ⊗ 1_{n}.

Assume now that B has the cancellation property. To establish the isomorphism (4), note first that n([p_{j}]_{0} − [e]_{0}) = 0 by (i). Retaining the notation from Example 3.5, we define a group homomorphism φ: G_{n} → H by φ(1, 0) = [e]_{0} and φ(0, e_{j}) = [p_{j}]_{0}−[e]_{0}. φ is clearly surjective. For any (k, t) ∈ G_{n} with t = ∑_{j=1}^{N} r_{j}e_{j}, 0 ≤ r_{j} < n,

φ(k, t) = k[e]_{0} + ∑_{j=1}^{N} r_{j}([p_{j}]_{0} − [e]_{0}) = (k − d(t))[e]_{0} + ∑_{j=1}^{N} r_{j}[p_{j}]_{0}.

It follows that φ(k, t) ≥ 0 if (k, t) ≥ 0. Conversely, if (k, t) is not positive, then k − d(t) ≤ −1, and so

φ(k, t) = (k − d(t))[e]_{0} + ∑_{j=1}^{N} r_{j}[p_{j}]_{0} ≤ (n − 1)([p_{1}]_{0} + [p_{2}]_{0} + ⋯ + [p_{N}]_{0}) − [e]_{0}.

By (ii) and the assumption that B has the cancellation property, the element on the right-hand side of this inequality is not positive. All in all we have shown that φ(k, t) ≥ 0 if and only if (k, t) ≥ 0. This entails that φ is injective and that φ(G_{n}^{+}) = H ∩ K_{0}(B)^{+}.

Since {q_{j}}_{j=1}^{∞} is an approximate unit for B, an element g ∈ K_{0}(B) lies in Σ(B) if and only if 0 ≤ g ≤ [q_{j}]_{0} for some j. Notice that φ(h_{j}) = [q_{j}]_{0}. Hence φ(k, t) ∈ Σ(B) if and only if 0 ≤ (k, t) ≤ h_{j} for some j, and this shows that φ(Σ_{n}) = H ∩ Σ(B). □

Remark 3.7 Corollary 4.2 and Proposition 5.2 contain for each prime number n examples of C^{∗}-algebras with projections e, p_{1}, p_{2}, p_{3}, … satisfying (i) and (ii) of Proposition 3.6. The C^{∗}-algebras in Proposition 5.2 have the cancellation property (being of stable rank one).

Remark 3.8 One can replace condition (i) in Proposition 3.6 by a weaker condition such as for example e ≾ p_{j} ⊗ 1_{n} for all j, and still obtain that the C^{∗}-algebra B defined in (3) has the property that M_{k}(B) is not stable for 1 ≤ k < n and M_{n}(B) is stable. However, with this weaker condition one would not have a description of the scaled ordered group as in (4).

4 The commutative case

We realize for each positive prime number n projections e, p_{1}, p_{2}, p_{3}, … satisfying conditions (i) and (ii) of Proposition 3.6, with respect to that n, inside a C^{∗}-algebra which is stably isomorphic to a commutative C^{∗}-algebra. At the same time, Lemma 4.1 below, is a key ingredient in Section 5.

If π: X_{1} → X_{2} is a continuous function, then π^{∗} will denote the map from the cohomology groups of X_{2} to the cohomology groups of X_{1}, and the same symbol will be used to denote the map from vector bundles over X_{2} to vector bundles over X_{1}. By naturality of the Euler class, e(π^{∗}(ξ)) = π^{∗}(e(ξ)) for all complex vector bundles ξ over Y.

The proof of Lemma 4.1 below is almost identical to the proof of [6, Theorem 3.4]. The statements of Lemma 4.1 and of [6, Theorem 3.4] are, however, quite different. Therefore, and for the convenience of the reader, we include a proof of Lemma 4.1.

Let 𝔻 denote the unit disk in the complex plane. Consider for each integer n ≥ 2 the equivalence relation ∼ on 𝔻 given by: z ∼ w if z = w or if |z| = |w| = 1 and z^{n} = w^{n}. Put Y_{n} = 𝔻/∼.

Lemma 4.1 Let n be a positive prime number, and put X = Y_{n}^{n−1}. There exists a complex line bundle ω over X with the following properties. Let m be a positive integer, let π_{1}, π_{2}, …, π_{m}: X^{m} → X be the coordinate maps, and set

ξ_{k}^{(m)} = π_{1}^{∗}(ω) ⊕ π_{2}^{∗}(ω) ⊕ ⋯ ⊕ π_{k}^{∗}(ω), 1 ≤ k ≤ m,

which is a complex vector bundle over X^{m} of dimension k. Let θ_{d} denote the trivial complex vector bundle (over X or X^{m}) of (complex) dimension d. Then

(i) nω ≅ θ_{n},

(ii) if (n − 1)ξ_{k}^{(m)} ⊕ θ_{d_{1}} ≅ η ⊕ θ_{d_{2}} for some complex vector bundle η over X^{m}, and some positive integers d_{1} and d_{2}, then d_{1} ≥ d_{2}, and

(iii) ω ⊕ η ≅ θ_{n} for some (n − 1)-dimensional complex vector bundle η over X.

Proof: Recall that H^{2}(Y_{n}; ℤ) ≅ ℤ/nℤ. There is a complex line bundle ζ over Y_{n} with non-trivial Euler class e(ζ) ∈ H^{2}(Y_{n}; ℤ), and with nζ ≅ θ_{n}. Let ν_{1}, ν_{2}, …, ν_{n−1} : X = Y_{n}^{n−1} → Y_{n} be the coordinate projections, and set

ω = ν_{1}^{∗}(ζ) ⊗ ν_{2}^{∗}(ζ) ⊗ ⋯ ⊗ ν_{n−1}^{∗}(ζ).

Then ω is a complex line bundle over X, and successive applications of the isomorphism nζ ≅ θ_{n} = nθ_{1}, yield nω ≅ θ_{n}. Hence (i) holds, and (iii) is a trivial consequence of (i).

To prove claim (ii) we first show that the Euler class, e((n − 1)ξ_{k}^{(m)}), is non-zero. The Euler class of ω is given by

e(ω) = ∑_{j=1}^{n−1} ν_{j}^{∗}(e(ζ)),  (5)

cf. [4, Proposition V.3.10]. By the product formula for the Euler class, cf. [4, Proposition V.3.10],

e((n − 1)ξ_{k}^{(m)}) = ∏_{j=1}^{k} π_{j}^{∗}(e(ω)^{n−1}).  (6)

Since e(ζ)^{2} ∈ H^{4}(Y_{n}; ℤ) and H^{4}(Y_{n}; ℤ) = 0, it follows from (5) and (6) that

e(ω)^{n−1} = (n − 1)! ∏_{i=1}^{n−1} ν_{i}^{∗}(e(ζ)).

Let ρ_{1}, ρ_{2}, …, ρ_{k}: X^{k} → X and π: X^{m} → X^{k} be the projections maps. Then π_{j} = ρ_{j} ∘ π, and π^{∗}: H^{2k}(X^{k}; ℤ) → H^{2k}(X^{m}; ℤ) is an injection. The map

μ: H^{2}(Y_{n}; ℤ) ⊗ H^{2}(Y_{n}; ℤ) ⊗ ⋯ ⊗ H^{2}(Y_{n}; ℤ) → H^{2k(n−1)}(X^{k}; ℤ)

given by

μ(x_{1,1} ⊗ x_{1,2} ⊗ ⋯ ⊗ x_{k,n−1}) = ∏_{j=1}^{k} ∏_{i=1}^{n−1}(ρ_{j}^{∗} ∘ ν_{i}^{∗})(x_{i,j}),

is injective by the Künneth formula. Now,

e((n − 1)ξ_{k}^{(m)}) = ∏_{j=1}^{k} π_{j}^{∗}(e(ω)^{n−1})

= ∏_{j=1}^{k} π_{j}^{∗}((n − 1)! ∏_{i=1}^{n−1} ν_{i}^{∗}(e(ζ)))

= (n − 1)!^{k}π^{∗}( ∏_{j=1}^{k} ∏_{i=1}^{n−1}(ρ_{j}^{∗} ∘ ν_{i}^{∗})(e(ζ))
= (π^{∗} ∘ μ)((n − 1)!^{k}e(ζ) ⊗ e(ζ) ⊗ ⋯ ⊗ e(ζ)).

The element e(ζ) ⊗ e(ζ) ⊗ ⋯ ⊗ e(ζ) has order n in H^{2}(Y_{n}; ℤ) ⊗ H^{2}(Y_{n}; ℤ) ⊗ ⋯ ⊗ H^{2}(Y_{n}; ℤ). Because n is assumed to be prime, and because π^{∗} ∘ μ is injective, we get that e((n − 1)ξ_{k}^{(m)}) ≠ 0.

Assume (ii) were false. Then (n − 1)ξ_{k}^{(m)} ⊕ θ_{d_{1}} ≅ η ⊕ θ_{d_{2}} for some η and some positive integers d_{1} < d_{2}. Hence (n − 1)ξ_{k}^{(m)} would be stably isomorphic to η ⊕ θ_{d_{2}−d_{1}}. The Euler class is invariant under stable isomorphism, and the Euler class of a trivial bundle (of dimension ≥ 1) is zero, and so by the product formula we get e((n − 1)ξ_{k}^{(m)}) = 0, a contradiction. □

George Elliott pointed out to me that one obtains the following corollary from Lemma 4.1:

Corollary 4.2 Let n be a positive prime number, let Z be the infinite Cartesian product of Y_{n} with itself. Then there exist projections e, p_{1}, p_{2}, p_{3}, … in M_{n}(C(Z)) satisfying

(i) e ⊗ 1_{n} ∼ p_{j} ⊗ 1_{n} for all j,

(ii) e is not equivalent to a subprojection of (p_{1} ⊕p_{2} ⊕⋯⊕p_{j})⊗1_{n−1} for any j ≥ 1.

Proof: Let ω be the complex line bundle over X = Y_{n}^{n−1} from Lemma 4.1 and use Lemma 4.1 (iii) to find a projection p ∈ C(X, M_{n}(ℂ)) = M_{n}(C(X)) that corresponds to ω. Identify Z with ∏_{j=1}^{∞} X, and let π_{j}: Z → X, j ∈ ℕ, be the coordinate maps. Put p_{j} = p ∘ π_{j} ∈ C(Z, M_{n}(ℂ)) = M_{n}(C(Z)), and let e ∈ M_{n}(C(Z)) be a onedimensional constant projection. It follows from Lemma 4.1 (i) that p_{j} ⊗ 1_{n} ∼ e ⊗ 1_{n} for all j. To see (ii), view M_{n}(C(Z)) as the inductive limit,

M_{n}(C(X)) → M_{n}(C(X^{2})) → M_{n}(C(X^{3})) → ⋯ → M_{n}(C(Z)),

so that e, p_{1}, p_{2}, …, p_{j} ∈ M_{n}(C(X^{j})). Then, by Lemma 4.1 (ii), for each k and for each m ≥ k, e is not equivalent to a subprojection of (p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{k}) ⊗ 1_{n−1} in (a matrix algebra over) M_{n}(C(X^{m})). This implies that (ii) holds. □

Combining Corollary 4.2 with Proposition 3.6 we get for each prime number n a hereditary sub-C^{∗}-algebra B of C(Z)⊗K such that M_{k}(B) is not stable for 1 ≤ k < n, and M_{n}(B) is stable. Proceeding as in the proof of Theorem 5.3 one can find such examples B for all integers n ≥ 2.

5 The simple case

We use an inductive limit construction, like the one Villadsen used in [7], to obtain projections as in Proposition 3.6 inside a simple C^{∗}-algebra.

Fix a positive prime number n. Let {k_{j}}_{j=1}^{∞} be a sequence of positive integers chosen large enough so that

∑_{j=1}^{∞} (1 − ∏_{i=j}^{∞} \frac{k_{i}}{1 + k_{i}} ) < \frac{1}{n − 1}.  (7)

Define inductively another sequence of integers {m_{j}}_{j=1}^{∞} by m_{1} = 1 and m_{j+1} = m_{j}(k_{j} + 1).

Let Y_{n} = 𝔻/∼ be as defined in Section 4, and put X = Y_{n}^{n−1}. Define inductively a sequence of spaces {X_{j}}_{j=1}^{∞} by setting X_{1} = X and X_{j+1} = X_{j}^{k_{j}} × X^{m_{j+1}}. Set

A_{j} = M_{2^{n−1}m_{j}}(C(X_{j})) = C(X_{j}, M_{2^{n−1}m_{j}}(ℂ)).

Choose x_{j} ∈ X_{j} appropriately (in a way which will be made precise later), and define ∗-homomorphisms φ_{j}: A_{j} → A_{j+1} by

φ_{j}(f)(x) = diag((f ∘π_{1}^{j})(x), (f ∘π_{2}^{j})(x), …, (f ∘π_{k_{j}}^{j})(x), f(x_{j})), x ∈ X_{j+1}, f ∈ A_{j},

where π_{1}^{j}, π_{2}^{j}, …, π_{k_{j}}^{j}: X_{j+1} = X_{j}^{k_{j}} × X^{m_{j+1}} → X_{j} are the projections from the first factor of X_{j+1}.

Let (A, μ_{j}: A_{j} → A) be the inductive limit of the sequence

\xymatrix{A_{1} \ar[r]^{φ_{1}} & A_{2} \ar[r]^{φ_{2}} & A_{3} \ar[r]^{φ_{3}} & ⋯} .

It will be convenient to have an expression for the composed connecting maps φ_{i,j}: A_{j} → A_{i} for i > j. For this purpose set

k_{i,j} = ∏_{n=j}^{i−1} k_{n},  l_{i,j} = ∏_{n=j}^{i−1}(k_{n} + 1) − ∏_{n=j}^{i−1} k_{n},  m_{i,j} = ∑_{n=j+1}^{i} m_{n}k_{i,n}, (8)

(with the convention that k_{i,i} = 1). Then X_{i} = X_{j}^{k_{i,j}} × X^{m_{i,j}}, and the composed connecting maps are up to unitary equivalence given by

φ_{i,j}(f)(x)

= diag((f ∘ π_{1}^{i,j})(x), (f ∘ π_{2}^{i,j})(x), …, (f ∘ π_{k_{i,j}}^{i,j})(x), f(x_{1}^{i,j}), f(x_{2}^{i,j}), …, f(x_{l_{i,j}}^{i,j})).

The maps π_{1}^{i,j}, π_{2}^{i,j}, …, π_{k_{i,j}}^{i,j}: X_{i} = X_{j}^{k_{i,j}} × X^{m_{i,j}} → X_{j} are here the projections onto the first k_{i,j} coordinates of X_{i}, the set

X_{j}^{i} := {x_{1}^{i,j}, x_{2}^{i,j}, …, x_{l_{i,j}}^{i,j}} ⊆ X_{j}

is for i ≥ j + 2 equal to X_{j}^{i−1} ∪ {π_{1}^{i,j}(x_{i}), π_{2}^{i,j}(x_{i}), …, π_{k_{i,j}}^{i,j}(x_{i})}, where each element of the first set is repeated k_{i} + 1 times, and X_{j}^{j+1} = {x_{j}}.

Choose the points x_{j} ∈ X_{j} such that ⋃_{r=j+1}^{∞} X_{j}^{r} is dense in X_{j} for each j ∈ ℕ. Since each X_{j}^{i} is finite and since no X_{j} has isolated points this will entail that ⋃_{r=i}^{∞} X_{j}^{r} is dense in X_{j} for each j ∈ ℕ and for every i > j.

By [2, Proposition 1] and [7, Proposition 10] we get:

Proposition 5.1 The C^{∗}-algebra A is simple and has stable rank one.

With the C^{∗}-algebra A and the prime number n as above, we have:

Proposition 5.2 There exist projections e, p_{1}, p_{2}, p_{3}, … in A so that

(i) p_{j} ⊗ 1_{n} ∼ e ⊗ 1_{n} for all j ≥ 1, and

(ii) e is not equivalent to a subprojection of (p_{1} ⊕p_{2} ⊕⋯⊕p_{j})⊗1_{n−1} for any j ≥ 1.

Proof: By Lemma 4.1 (iii) there exists a projection q ∈ A_{1} = M_{2^{n−1}}(C(X)) which corresponds to the complex line bundle ω. Let ρ_{1}, ρ_{2}, …, ρ_{m_{j}}: X_{j} = X_{j−1}^{k_{j−1}} × X^{m_{j}} → X be coordinate projections corresponding to the last factor of X_{j}. Set q_{1} = q, set

q_{j} = diag(q ∘ ρ_{1}, q ∘ ρ_{2}, …, q ∘ ρ_{m_{j}}) ∈ A_{j},

for j ≥ 2, and set p_{j} = μ_{j}(q_{j}) ∈ A. Let e_{1} ∈ A_{1} be a constant projection of dimension 1, so that e_{1} corresponds to the trivial complex line bundle θ_{1}, and set e = μ_{1}(e_{1}) ∈ A.

It follows from Lemma 4.1 (i) that q ⊗ 1_{n} ∼ e_{1} ⊗ 1_{n}. This implies that q_{j} ⊗ 1_{n} is equivalent to a constant projection. Since φ_{j,1}(e_{1}) ⊗ 1_{n} is a constant projection (in M_{n}(A_{j})) of the same dimension as q_{j} ⊗ 1_{n}, we find that q_{j} ⊗ 1_{n} ∼ φ_{j,1}(e_{1}) ⊗ 1_{n} in M_{n}(A_{j}). Hence

p_{j} ⊗ 1_{n} = μ_{j}(q_{j} ⊗ 1_{n}) ∼ μ_{j}(φ_{j,1}(e_{1}) ⊗ 1_{n}) = e ⊗ 1_{n}

in M_{n}(A).

For i ≥ j, put

f_{i,j} = φ_{i,1}(q_{1}) ⊕ φ_{i,2}(q_{2}) ⊕ ⋯ ⊕ φ_{i,j}(q_{j}).

Then p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{j} = μ_{i}(f_{i,j}), and f_{i,j} = φ_{i,j}(f_{j,j}). Observe that X_{j} = X^{d_{j}}, where d_{1} = 1 and d_{j+1} = n_{j}k_{j} +m_{j+1}. By inspection of the formula for the composed connecting maps φ_{j,l}, we find that the projection f_{j,j} corresponds to the vector bundle ξ_{d_{j}}^{(d_{j})} ⊕ θ_{c_{j}}, where c_{j} = ∑_{r=1}^{j} m_{r}l_{j,r}, cf. (8). From this we get that the projection

f_{i,j} corresponds to the vector bundle ξ_{a_{i,j}}^{(d_{i})} ⊕ θ_{b_{i,j}} over X_{i}, where a_{i,j} = k_{i,j}d_{j} and b_{i,j} = ∑_{r=1}^{j} m_{r}l_{i,r}, possibly after a permutation of the coordinates of X_{i}.

The trivial projection φ_{i,1}(e_{1}) has dimension m_{i} and corresponds therefore to the trivial vector bundle θ_{m_{i}}. Now,

\frac{1}{m_{i}} b_{i,j} = \frac{1}{m_{i}} ∑_{r=1}^{j} m_{r}l_{i,r}

= \frac{1}{m_{i}} ∑_{r=1}^{j} ∏_{s=1}^{r−1}(1 + k_{s})( ∏_{s=r}^{i−1}(1 + k_{s}) − ∏_{s=r}^{i−1} k_{s})

= ∑_{r=1}^{j} (1 − ∏_{s=r}^{i−1} \frac{k_{s}}{1 + k_{s}} )

≤ ∑_{r=1}^{∞} (1 − ∏_{s=r}^{∞} \frac{k_{s}}{1 + k_{s}} ) < \frac{1}{n − 1},

where the last inequality follows from (5). This shows that (n − 1)b_{i,j} < m_{i}. By Lemma 4.1 (ii), there exists no vector bundle η over X_{i} such that

η ⊕ θ_{m_{i}} ≅ (n − 1)ξ_{a_{i,j}}^{(d_{i})} ⊕ θ_{(n−1)b_{i,j}} (= (n − 1)(ξ_{a_{i,j}}^{(d_{i})} ⊕ θ_{b_{i,j}})),

or, equivalently, φ_{i,1}(e_{1}) is not equivalent to a subprojection of f_{i,j} ⊗ 1_{n−1}. Since this holds for all i > j, e is not equivalent to a subprojection of (p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{j}) ⊗ 1_{n−1}, and this completes the proof. □

Theorem 5.3 For each integer n ≥ 2 there exists a C^{∗}-algebra B such that M_{n}(B) is stable, and M_{k}(B) is not stable for 1 ≤ k < n. Moreover, B can be chosen to be simple, nuclear, with stable rank one and with an approximate unit consisting of projections.

Proof: Consider first the case where n is prime. Let B be the C^{∗}-algebra defined in display (3) in Proposition 3.6 corresponding to the C^{∗}-algebra A and to the projections e, p_{1}, p_{2}, p_{3}, … found in Proposition 5.2. Then B is a hereditary subalgebra of A⊗K, and since A is simple, nuclear and has stable rank one, it follows that B also has these properties (see [5, Theorem 3.3] for the last claim). The sequence {q_{j}}_{j=1}^{∞} is an approximate unit for B. By Proposition 3.6, M_{k}(B) is not stable for 1 ≤ k < n and M_{n}(B) is stable.

Suppose now that n ≥ 2 is an arbitrary integer. Observe that all integers ≥ (n − 1)^{2} belong to the set

⋃_{m=1}^{∞}((n − 1)m, nm].

Choose a prime number p ≥ (n − 1)^{2}. Then there exists an integer m ≥ 1 so that (n − 1)m < p ≤ nm. By the first part of the proof there exists a C^{∗}-algebra D with M_{p}(D) stable and M_{k}(D) not stable for 1 ≤ k < p. Set B = M_{m}(D). Then B is simple, nuclear, and has stable rank one and an approximate unit consisting of projections because D has these properties. Moreover, M_{k}(B) = M_{km}(D), and so, by Proposition 2.1, M_{k}(B) is stable if and only if km ≥ p, which, by the choice of p and m, happens if and only if k ≥ n. □

References

[1] B. Blackadar, K-theory for operator algebras, M. S. R. I. Monographs, vol. 5, Springer Verlag, Berlin and New York, 1986.

[2] M. Dădărlat, G. Nagy, A. Némethi, and C. Pasnicu, Reduction of topological stable rank in inductive limits of C^{∗}-algebras, Pacific J. Math. 153 (1992), 267–276.

[3] J. Hjelmborg and M. Rørdam, On stability of C^{∗}-algebras, J. Funct. Anal., to appear.

[4] M. Karoubi, K-theory, Grundlehren der mathematisches Wissenschaften, no. 226, Springer Verlag, 1978.

[5] M.A. Rieffel, Dimension and stable rank in the K-theory of C^{∗}-algebras, Proc. London Math. Soc. 46 (1983), no. (3), 301–333.

[6] M. Rørdam and J. Villadsen, On the ordered K_{0}-group of universal free product C^{∗}-algebras, preprint.

[7] J. Villadsen, Simple C^{∗}-algebras with perforation, J. Funct. Anal., to appear.

Mikael Rørdam

Department of Mathematics University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø, Denmark rordam@math.ku.dk