New York Journal of Mathematics

New York J. Math. 6 (2000) 285–306.

KK-Equivalence and Continuous Bundles of C^{∗}-Algebras

Klaus Thomsen

Abstract. A structural description is given of the separable C^{∗}-algebras that are contractible in KK-theory or E-theory. Subsequently it is shown that two separable C^{∗}-algebras, A and B, are KK-equivalent if and only if there is a bundle of separable C^{∗}-algebras over [0, 1] which is piecewise trivial with no more than six points of non-triviality such that the kernel of each fiber map is KK-contractible, all fiber maps are semi-split and such that the fibers at the endpoints of the interval are A and B.

Contents

1. Introduction 285 2. Locally contractible C^{∗}-algebras and triviality in E-theory 287 3. Semi-contractible C^{∗}-algebras and triviality in KK-theory 289 4. The structure of semi-contractible C^{∗}-algebras 290 5. Generalized inductive limits and continuous bundles 292 6. From KK-equivalence to continuous bundles 296 7. Concatenation of bundles 303 References 305

1. Introduction

Through the work directed at classifiying C^{∗}-algebras it has become apparent that the KK-theory of Kasparov, including the E-theory of Connes and Higson, offers much more than a convenient setup for dealing with topological K-theory. Although KK-theory was first invented and developed, from the late seventies through the eighties, as a tool to attack topological questions (the Novikov conjecture) and calculate the K-theory of group C^{∗}-algebras (the Baum-Connes conjecture), it has found some of its most profound applications in the classification program of the nineties. And although it is clear that KK-theory as a carrier of information about the structure of C^{∗}-algebras can not in general stand alone when we seek to classify

285^{ISSN 1076-9803/00}

Received June 15, 2000.

Mathematics Subject Classification. 19K35, 46L35.

Key words and phrases. C^{∗}-algebras, KK-theory, continuous fields.

C^{∗}-algebras up to isomorphism, it must necessarily play a major role in such efforts. As a consequence of this understanding it has become important to decide in which way the structures of two C^{∗}-algebras are related when they are KK-equivalent and/or equivalent in E-theory. Dadarlat has obtained one answer to this question in [D]: Two separable C^{∗}-algebras are equivalent in E-theory if and only if their stable suspensions are shape equivalent. The purpose of the present paper is to use recent results of the author on KK-theory and E-theory to give an alternative answer, and this time for KK-theory.

First of all we must decide what it means for a C^{∗}-algebra to be KK-contractible, i.e., KK-equivalent to 0. We do this first for E-theory in Section 2 and then modify the approach to handle KK-theory in Section 3. The central notion in the description of which C^{∗}-algebras are KK-contractible is called semi-contractibility. A C^{∗}-algebra is semi-contractible when the identity map of the algebra can be connected to 0 by a continuous path of completely positive contractions such that the maps in the path are almost multiplicative up to an arbitrary small toleration on any given finite subset. It turns out that a separable and stable C^{∗}-algebra A is KK-contractible if and only it is the quotient of a semi-contractible C^{∗}-algebra by a semi-contractible ideal. In order to identify semi-contractible C^{∗}-algebras in later parts of the paper we give a description of them involving generalized inductive limits in Section 4. The notion of a generalized inductive system of C^{∗}-algebras was introduced by Blackadar and Kirchberg in [BK] and the notion is a cornerstone in the approach here. The idea behind such systems comes clearly from the approximate intertwining of Elliott, [E], but it is also inspired by the E-theory of Connes and Higson, and it is only natural that we can use it in Section 5 to transfer KK-theory information, encoded in two completely positive asymptotic homomorphisms, into an isomorphism between two C^{∗}-algebras which are closely related to the two given KK-equivalent separable C^{∗}-algebras, A and B. The result is that we obtain a continuous bundle of C^{∗}-algebras which connects SA ⊗ K to SB ⊗ K and has several quite special properties.^{1}

In the last section we glue the bundle from Section 5 together with other bundles, notably a bundle considered by Elliott, Natsume and Nest in [ENN], to obtain our main result which says that two separable C^{∗}-algebras, A and B, are KK-equivalent if and only if they can be connected by a bundle of C^{∗}-algebras over [0, 1] of a particular form. Specifically, the bundle is a piecewise trivial bundle with no more than six points of non-triviality such that the kernel of each fiber map is KKcontractible, all fiber maps are semi-split and such that the fibers at the endpoints of the interval are A and B. In particular, it follows that all the fiber algebras are KK-equivalent to the bundle C^{∗}-algebra. As an immediate corollary we get that A and B are KK-equivalent if and only if there is a separable C^{∗}-algebra D and two surjective ∗-homomorphisms φ : D → A and ψ : D → B, both of which admit a completely positive section and have KK-contractible kernels. Any KK-equivalence between separable C^{∗}-algebras can therefore be realized by the Kasparov product of a surjective ∗-homomorphism with the inverse of a surjective ∗-homomorphism.

1I have chosen to follow the lead from [KW] and use the word ’bundle’ instead of ’field’.

2. Locally contractible C^{∗}-algebras and triviality in E-theory

Definition 2.1. A C^{∗}-algebra D is called locally contractible when the following holds: For every finite set F ⊆ D and every ϵ > 0 there is a pointwise norm continuous family of homogeneous maps, δ_{s} : D → D, s ∈ [0, 1], such that δ_{0} = 0, δ_{1} = id_{D}, ∥δ_{s}(a^{∗}) − δ_{s}(a)^{∗}∥ < ϵ, ∥δ_{s}(a) + δ_{s}(b) − δ_{s}(a + b)∥ < ϵ, ∥δ_{s}(a)δ_{s}(b) − δ_{s}(ab)∥ < ϵ and ∥δ_{s}(a)∥ < ∥a∥ + ϵ for all s ∈ [0, 1] and all a, b ∈ F.

Recall from [MT] that an extension of C^{∗}-algebras

\xymatrix{0 \ar[r] & J \ar[r] & E \ar[r]^{p} & A \ar[r] & 0}

is asymptotically split when there is an asymptotic homomorphism π_{t} : A → E, t ∈ [1, ∞), such that p ∘ π_{t} = id_{A} for all t. If one can choose π = {π_{t}}_{t∈[1,∞)} to be a completely positive asymptotic homomorphism, we say that the extension is completely positive asymptotically split.

Theorem 2.2. Let A be a separable C^{∗}-algebra. Then the following conditions are equivalent:

a) A is contractible in E-theory (i.e., [[SA ⊗ K, SA ⊗ K]] = 0).

b) The canonical extension

\xymatrix{0 \ar[r] & S^{2}A ⊗ K \ar[r] & cone(SA ⊗ K) \ar[r] & SA ⊗ K \ar[r] & 0}

is asymptotically split.

c) SA ⊗ K is locally contractible.

d) There is an extension

\xymatrix{0 \ar[r] & J \ar[r] & E \ar[r] & A ⊗ K \ar[r] & 0} (1)

of separable C^{∗}-algebras where J is locally contractible and E is contractible.

e) There is an extension (1) of separable C^{∗}-algebras where J and E are locally contractible.

For the proof of this we need to go from information about discrete asymptotic homomorphisms to information about genuine asymptotic homomorphisms. Let A and B be arbitrary separable C^{∗}-algebras. As in [Th1] we denote by [[A, B]]_{ℕ} the homotopy classes of discrete asymptotic homomorphisms from A to B. The shift σ, given by σ(φ)_{n} = φ_{n+1}, defines an automorphism of the group [[SA, SB]]_{ℕ}. Let [[SA, SB]]_{ℕ}^{σ} denote the fixed point group of σ. By Lemma 5.6 of [Th1] there is a short exact sequence

\xymatrix{0 \ar[r] & [[SA, SB]]^{0} \ar[r] & [[SA, SB]] \ar[r] & [[SA, SB]]_{ℕ}^{σ} \ar[r] & 0.} (2)

Here [[SA, SB]]^{0} is the subgroup of [[SA, SB]] consisting of the elements which can be represented by an asymptotic homomorphism φ = {φ_{t}} which is sequentially trivial in the sense that the sequence φ_{1}, φ_{2}, φ_{3}, … converges pointwise to zero, i.e., lim_{n→∞} φ_{n}(x) = 0 for all x ∈ SA. The surjective map [[SA, SB]] → [[SA, SB]]_{ℕ}^{σ} is obtained by restricting the parameters of the asymptotic homomorphisms from ℝ to ℕ.

Besides the extension (2) from [Th1] we need the observation that the composition product of two elements from [[−, −]]^{0} is always zero:

Lemma 2.3. Let ψ = {ψ}_{t∈ℝ} : B → C and φ = {φ_{t}}_{t∈ℝ} : A → B be asymptotic homomorphisms which are sequentially trivial, i.e., satisfy that lim_{n→∞} ψ_{n}(b) = 0, b ∈ B, and lim_{n→∞} φ_{n}(a) = 0, a ∈ A. It follows that [ψ] • [φ] = 0 in [[A, C]].

Proof. Choose equicontinuous sequentially trivial asymptotic homomorphisms ψ′ : B → C and φ′ : A → B such that lim_{t→∞} ∥ψ_{t}(b) − ψ′_{t}(b)∥ = 0, b ∈ B, and lim_{t→∞} ∥φ_{t}(a) − φ′_{t}(a)∥ = 0, a ∈ A. Let D be a countable dense subset of A. By definition of • there is a parametrization r : [1, ∞) → [1, ∞) such that [ψ] • [φ] is represented by any equicontinuous asymptotic homomorphism λ which satisfies that lim_{t→∞} λ_{t}(a) − ψ′_{s(t)} ∘ φ′_{t}(a) = 0, a ∈ D, for some parametrization s ≥ r. We leave the reader to construct a parametrization s ≥ r with the property that s(t) ∈ ℕ for all t outside a neighbourhood of ℕ ⊆ [1, ∞) and such that lim_{t→∞} ψ′_{s(t)} ∘ φ′_{t}(a) = 0 for all a ∈ D. By equicontinuity of λ this implies that lim_{t→∞} λ_{t}(a) = 0 for all a ∈ A. □

We can now give the proof of Theorem 2.2.

Proof. a) ⇒ b): Set B = A ⊗ K. Since [id_{SB}] = 0 in [[SB, SB]], it follows from Theorem 1.1 of [Th2] that there is an asymptotic homomorphism μ = {μ_{t}}_{t∈[1,∞)} : cone(B) → SB and a norm continuous path, U_{t}, t ∈ [1, ∞), of unitaries in M_{2}(SB)^{+} such that

lim_{t→∞} (\begin{matrix}b \\ & μ_{t}(b) \end{matrix}) − U_{t} (\begin{matrix}0 \\ & μ_{t}(b) \end{matrix}) U_{t}^{∗} = 0

for all b ∈ SB. Let V_{1}, V_{2} be isometries in the multiplier algebra M(SB) of SB such that V_{1}V_{1}^{∗} + V_{2}V_{2}^{∗} = 1 and consider a strictly continuous path W_{s}, s ∈ [0, 1], of isometries in M(SB) such that W_{0} = 1 and W_{1} = V_{1}. Let T_{t} be the image of U_{t} under the isomorphism M_{2}(M(SB)) → M(SB) obtained from V_{1}, V_{2}. Let ψ_{s} : cone(B) → cone(B), s ∈ [0, 1], be the canonical trivialization of cone(B), i.e., ψ_{s}(f)(t) = f(st). Define δ_{s}^{t} : SB → SB, s ∈ [0, 4], by

δ_{s}^{t}(a) = \begin{cases}W_{s}aW_{s}^{∗}, & s ∈ [0, 1] \\ V_{1}aV_{1}^{∗} + V_{2}μ_{t}(ψ_{s−1}(a))V_{2}^{∗}, & s ∈ [1, 2] \\ (3 − s)[V_{1}aV_{1}^{∗} + V_{2}μ_{t}(a)V_{2}^{∗}] + (s − 2)T_{t}V_{2}μ_{t}(a)V_{2}^{∗}T_{t}^{∗}, & s ∈ [2, 3] \\ T_{t}V_{2}μ_{t}(ψ_{4−s}(a))V_{2}^{∗}T_{t}^{∗}, & s ∈ [3, 4] \end{cases}

Define π_{t} : SB → cone(SB) by π_{t}(x)(s) = δ_{4−4s}^{t}(x).

b) ⇒ c): Let π = {π_{t}}_{t∈[1,∞)} : SA ⊗ K → cone(SA ⊗ K) be an asymptotic homomorphism such that π_{t}(x)(1) = x for all t and all x. Fix a t ≥ 1. If a finite subset F ⊆ SA ⊗ K and ϵ > 0 are given, define δ_{s} : SA ⊗ K → SA ⊗ K by δ_{s}(x) = π_{t}(x)(s). If t is large enough {δ_{s}}_{s∈[0,1]} will meet the requirements of Definition 2.1.

c) ⇒ d): The canonical extension

\xymatrix{0 \ar[r] & SA ⊗ K \ar[r] & cone(A ⊗ K) \ar[r] & A ⊗ K \ar[r] & 0}

has the stated properties.

d) ⇒ e): This is trivial.

e) ⇒ a) : Thanks to excision in E-theory it suffices to show that a separable locally contractible C^{∗}-algebra D is contractible in E-theory. We first show that SD ⊗ K is locally contractible when D is. Let F_{1} ⊆ F_{2} ⊆ F_{3} ⊆ ⋯ be finite subsets with dense union in D. Since D is locally contractible we can construct homogeneous maps δ_{n} : D → cone(D) such that δ_{n}(d)(1) = d for all n ∈ ℕ, d ∈ D, and ∥δ^{n}(a)δ^{n}(b)−δ^{n}(ab)∥ ≤ \frac{1}{n}, ∥δ^{n}(a^{∗})−δ^{n}(a)^{∗}∥ ≤ \frac{1}{n}, ∥δ^{n}(a+b)−δ^{n}(a)−δ^{n}(b)∥ ≤ \frac{1}{n}, ∥δ^{n}(a) − δ^{n}(b)∥ ≤ ∥a − b∥ + \frac{1}{n} for all a, b ∈ F_{n}. The sequence \{δ^{n}\} defines a ∗-homomorphism Δ : D → ∏_{n} cone(D)/ ⊕_{n} cone(D) in the obvious way. By the Bartle-Graves selection theorem there is a continuous and homogeneous lift ψ : D → ∏_{n} cone(D) of Δ. Set ψ_{n}(d) = ψ(d)(n). Then {ψ_{n}} : D → cone(D) is an equicontinuous family of maps forming a discrete asymptotic homomorphism such that lim_{n→∞} ∥ψ_{n}(d)(1) − d∥ = 0 for all d ∈ D. The tensor product construction from [CH] gives us now an equicontinuous discrete asymptotic homomorphism ψ′ : D ⊗ SK → cone(D) ⊗ SK such that lim_{n→∞} ∥ψ′_{n}(d ⊗ x) − ψ_{n}(d) ⊗ x∥ = 0 for d ∈ D, x ∈ SK. In particular it follows that lim_{n→∞}(ev_{1} ⊗ id_{SK}) ∘ ψ′(z) = z when ev_{1} : cone(D) → D denotes evaluation at 1. It follows then readily that SD ⊗ K ≃ D ⊗ SK is locally contractible. Write SD ⊗ K = \overline{⋃_{n} F_{n}} where F_{1} ⊆ F_{2} ⊆ F_{3} ⊆ ⋯ are finite subsets. Since SD ⊗K is locally contractible we can construct a pointwise norm continuous path δ_{s}, s ∈ [1, ∞), of homogeneous maps δ_{s} : SD ⊗ K → SD ⊗ K such that δ_{n} = id_{SD⊗K} and δ_{n+ \frac{1}{2}} = 0 for all n ∈ ℕ, and ∥δ_{s}(a^{∗}) − δ_{s}(a)^{∗}∥ < \frac{1}{n}, ∥δ_{s}(a) + δ_{s}(b) − δ_{s}(a + b)∥ < \frac{1}{n}, ∥δ_{s}(a)δ_{s}(b) − δ_{s}(ab)∥ < \frac{1}{n}, ∥δ_{s}(a) − δ_{s}(b)∥ < ∥a − b∥ + \frac{1}{n}, s ∈ [n, n + 1], a, b ∈ F_{n}. There is then an asymptotic homomorphism \overline{δ} = \{\overline{δ}_{s}\}_{s∈[1,∞)} : SD ⊗ K → SD ⊗ K such that lim_{s→∞} ∥\overline{δ}_{s}(a) − δ_{s}(a)∥ = 0 for all a ∈ SD ⊗K. In particular, [\overline{δ}|_{ℕ}] = [id_{SD⊗K}] in [[SD ⊗K, SD ⊗K]]_{ℕ}^{σ}. It follows then from (2) that [\overline{δ}] − [id_{SD⊗K}] ∈ [[SD ⊗ K, SD ⊗ K]]^{0}. Since lim_{n→∞} \overline{δ}_{n+ \frac{1}{2}} (a) = 0 for all a ∈ SD ⊗ K we have also that [\overline{δ}|_{ℕ}] = 0 in [[SD ⊗ K, SD ⊗ K]]_{ℕ}^{σ}. Consequently [id_{SD⊗K}] ∈ [[SD⊗K, SD⊗K]]^{0} by (2) and hence [id_{SD⊗K}] = [id_{SD⊗K}]•[id_{SD⊗K}] = 0 by Lemma 2.3. □

It follows from Theorem 2.2 that the class of separable E-contractible C^{∗}- algebras is the least class of separable C^{∗}-algebras which contains the locally contractible C^{∗}-algebras and is closed under stabilization and under the formation of quotients A/I where both A and the ideal I are in the class.

3. Semi-contractible C^{∗}-algebras and triviality in KK-theory

Definition 3.1. A C^{∗}-algebra D is called semi-contractible when the following holds: For every finite set F ⊆ D and every ϵ > 0 there is a pointwise norm continuous family of completely positive contractions, δ_{s} : D → D, s ∈ [0, 1], such that δ_{0} = 0, δ_{1} = id_{D} and ∥δ_{s}(a)δ_{s}(b)−δ_{s}(ab)∥ < ϵ for all s ∈ [0, 1] and all a, b ∈ F.

The results of [Th1] and [Th2] which were used in the last section all have analogues for completely positive asymptotic homomorphisms which were also presented in [Th1] and [Th2]. It is therefore easy to use the same arguments to prove the following result.

Theorem 3.2. Let A be a separable C^{∗}-algebra. Then the following conditions are equivalent :

a) A is contractible in KK-theory (i.e., KK(A, A) = 0).

b) The canonical extension

\xymatrix{0 \ar[r] & S^{2}A ⊗ K \ar[r] & cone(SA ⊗ K) \ar[r] & SA ⊗ K \ar[r] & 0}

is completely positive asymptotically split.

c) SA ⊗ K is semi-contractible.

d) There is a semi-split extension

\xymatrix{0 \ar[r] & J \ar[r] & E \ar[r] & A ⊗ K \ar[r] & 0} (3)

of separable C^{∗}-algebras where J is semi-contractible and E is contractible.

e) There is a semi-split extension (3) of separable C^{∗}-algebras where J and E are semi-contractible.

It follows from Theorem 3.2 that the class of separable KK-contractible C^{∗}- algebras is the least class of separable C^{∗}-algebras containing the semi-contractible C^{∗}-algebras and closed under stabilization and under the formation of quotients A/I when both A and the ideal I are in the class, and there is a completely positive section for the quotient map A → A/I.

Clearly,

{contractible C^{∗}-algebras} ⊆ {semi-contractible C^{∗}-algebras}

⊆ {locally contractible C^{∗}-algebras}.

The following examples show that both inclusions are strict, also when we restrict attention to separable C^{∗}-algebras that are stable suspensions.

Example 3.3. We give here an example of a class of separable C^{∗}-algebras E which are KK-contractible, and whose stable suspension SE ⊗ K are not contractible.^{2} Let B be a unital separable infinite dimensional simple C^{∗}-algebra which is KKequivalent to an abelian C^{∗}-algebra A. The KK-equivalence is represented by a semi-split extension of SA by B ⊗ K as an element of Ext^{−1}(SA, B). By stabilizing and suspending the extension becomes

\xymatrix{0 \ar[r] & SB ⊗ K \ar[r] & SE ⊗ K \ar[r] & S^{2}A ⊗ K \ar[r] & 0.}

The algebra E is KK-contractible. This is because the connecting maps of the sixterm exact sequence arising by applying the functor KK(SE, −) to the extension is given by taking the Kasparov product with the KK-equivalence which the extension represents and hence are isomorphisms. Another way to see this is to use the UCTtheorem of Rosenberg and Schochet, [RS]. However, SE ⊗ K is not contractible because the properties of B ensure that Hom(SB, K) = 0 so that also Hom(SB ⊗ K, S^{2}A ⊗ K) = 0. Consequently, any ∗-endomorphism of SE ⊗ K must leave SB ⊗ K ⊆ SE ⊗ K globally invariant. So when SB ⊗ K is not contractible (as can easily be arranged by requiring K_{∗}(B) ≠ 0), SE ⊗ K will not be contractible.

Example 3.4. In [S] Skandalis gave an example of a separable C^{∗}-algebra A which is trivial in E-theory, but not in KK-theory. Hence by Theorems 3.2 and 2.2, SA⊗K is locally contractible, but not semi-contractible.

4. The structure of semi-contractible C^{∗}-algebras

To construct and study semi-contractible C^{∗}-algebras we need the notion of a generalized inductive system of C^{∗}-algebras and the inductive limit of a such a system. This was defined by Blackadar and Kirchberg in [BK] and we shall use their terminology and results. Given a contractible C^{∗}-algebra D, a trivialization of D will be a pointwise norm continuous path, ψ_{s}, s ∈ [0, 1], of endomorphisms of D such that ψ_{0} = 0 and ψ_{1} = id_{D}.

2I am grateful to Mikael Rørdam for pointing examples of this kind out to me.

Definition 4.1. A separable C^{∗}-algebra B is called approximately contractible when there is a sequence

\xymatrix{B_{1} \ar[r]^{φ_{1}} & B_{2} \ar[r]^{φ_{2}} & B_{3} \ar[r]^{φ_{3}} & ⋯} (4)

of contractible C^{∗}-algebras B_{n} with trivializations ψ_{s}^{n}, s ∈ [0, 1], and completely positive contractions φ_{n} : B_{n} → B_{n+1}, p_{n} : B_{n+1} → B_{n}, such that

1) for k ∈ ℕ, a, b ∈ B_{k} and ϵ > 0, there is a N ∈ ℕ such that

∥φ_{m,n}(ψ_{s}^{n} ∘ φ_{n,k}(a))φ_{m,n}(ψ_{s}^{n} ∘ φ_{n,k}(b)) − φ_{m,n}(ψ_{s}^{n}(φ_{n,k}(a)φ_{n,k}(b)))∥ < ϵ

for all s ∈ [0, 1] and all N ≤ n ≤ m,

2) p_{k+1} ∘ φ_{k+1} ∘ φ_{k} = φ_{k} for all k, and B ≃ \underrightarrow{lim}(B_{n}, φ_{n,k}).

Here φ_{n,k} is the composite map φ_{n−1} ∘⋯∘φ_{k} : B_{k} → B_{n} when n > k. (Observe that we use an index convention which the is the reverse of the one used in [BK].) Note that condition 1) of Definition 4.1 ensures that the sequence (4) is a generalized inductive system in the sense of Blackadar and Kirchberg, cf. Definition 2.1.1 of [BK].

Proposition 4.2. Let D be a separable C^{∗}-algebra. Then the following conditions are equivalent.

a) D is semi-contractible.

b) D ≃ \underrightarrow{lim}(cone(D), φ_{n,k}), where φ_{n} : cone(D) → cone(D) is a sequence of completely positive contractions such that 1) of Definition 4.1 holds relative to the canonical trivialization of cone(D), and there are completely positive contractions p_{k} : cone(D) → cone(D) such that 2) of Definition 4.1 holds.

c) D is approximately contractible.

Proof. a) ⇒ b) : Let ψ_{s}, s ∈ [0, 1], be the canonical trivialization of cone(D). Let F_{1} ⊆ F_{2} ⊆ F_{3} ⊆ ⋯ be a sequence of finite sets with dense union in cone(D). Since D is semi-contractible we can construct, recursively, a sequence δ^{n} of paths δ_{s}^{n} : D → D, s ∈ [0, 1], of completely positive contractions such that ∥δ_{s}^{n}(a)δ_{s}^{n}(b) − δ_{s}^{n}(ab)∥ < \frac{1}{n} for all s ∈ [0, 1] and all a, b ∈ \{δ_{s}^{j}(F_{n}) : s ∈ [0, 1], j < n\}. Define χ_{n} : D → cone(D) by χ_{n}(d)(s) = δ_{s}^{n}(d), s ∈ [0, 1], and μ : cone(D) → D by μ(f) = f(1). Set φ_{n} = χ_{n+1} ∘ μ and note that the diagram

[figure: D; χ_{1}; D; χ_{2}; D; χ_{3}; ⋯; cone(D); φ_{1}; μ; cone(D); φ_{2}; μ; cone(D); φ_{3}; μ; ⋯]

commutes. Since φ_{n,k} = χ_{n} ∘μ, n > k, it follows easily that (cone(D), φ_{n,k}) satisfies 1) of Definition 4.1 and the above diagram shows that D ≃ \underrightarrow{lim}(cone(D), φ_{n,k}), cf. [BK]. Define p_{k} : cone(B) → cone(B) by p_{k}(g)(s) = δ_{s}^{k}(g(1)). Then the p_{k}’s are completely positive contractions such that 2) of Definition 4.1 holds.

b) ⇒ c) : This is trivial.

c) ⇒ a) : Let D ≃ \underrightarrow{lim}(B_{n}, φ_{n,k}) where the B_{n}’s are contractible C^{∗}-algebras with trivializations ψ_{s}^{n}, s ∈ [0, 1], and completely positive contractions p_{k} : B_{k+1} → B_{k} such that 1) and 2) of Definition 4.1 hold. Set p_{n,m} = p_{n} ∘ p_{n−1} ∘ ⋯ ∘ p_{m−1} and let q : ∏_{i} B_{i} → ∏_{i} B_{i}/ ⊕_{i} B_{i} be the quotient map. Let x = (x_{1}, x_{2}, x_{3}, …) ∈ ∏_{i} B_{i} be an element such that q(x) ∈ \underrightarrow{lim}(B_{n}, φ_{n,k}). We assert that lim_{m→∞} p_{n,m}(x_{m}) exists in B_{n}. By an obvious \frac{ϵ}{2}-argument we may assume that q(x) = φ_{∞,l}(y) for some y ∈ B_{l}, l > n. In this case we see from 2) of Definition 4.1 that lim_{m→∞} p_{n,m}(x_{m}) = p_{n,l+1} ∘ φ_{l}(y). It follows that there is a completely positive contraction \overline{p}_{n} : \underrightarrow{lim}(B_{n}, φ_{n,k}) → B_{n} such that \overline{p}_{n} ∘ φ_{∞,l}(y) = p_{n,l+1} ∘ φ_{l}(y), y ∈ B_{l}, l > n. For l < n we find that

\overline{p}_{n} ∘ φ_{∞,l}(x) = lim_{m→∞} p_{n,m} ∘ φ_{m,l}(x) = φ_{n,l}(x), (5)

x ∈ B_{l}. It follows that lim_{n→∞} φ_{∞,n} ∘ \overline{p}_{n}(x) = x for all x ∈ \underrightarrow{lim}(B_{n}, φ_{n,k}). Furthermore, it follows from (5), the density of ⋃_{l} φ_{∞,l}(B_{l}) in \underrightarrow{lim}(B_{n}, φ_{n,k}) and 1) of Definition 4.1 that when F is a finite subset of \underrightarrow{lim}(B_{n}, φ_{n,k}) and ϵ > 0, then there is a n so large that δ_{s}^{n} = φ_{∞,n}∘ψ_{s}^{n}∘\overline{p}_{n}, s ∈ [0, 1], is a pointwise norm continuous path of completely positive contractions on \underrightarrow{lim}(B_{n}, φ_{n,k}) such that δ_{0}^{n} = 0, ∥δ_{1}^{n}(a)−a∥ < ϵ and ∥δ_{s}^{n}(ab) − δ_{s}^{n}(a)δ_{s}^{n}(b)∥ < ϵ for all a, b ∈ F. It follows that \underrightarrow{lim}(B_{n}, φ_{n,k}) is semicontractible. □

5. Generalized inductive limits and continuous bundles

Let A_{1}, A_{2}, A_{3}, … be a sequence of C^{∗}-algebras. For each n let φ_{t}^{n} : A_{n} → A_{n+1}, t ∈ [0, 1], be a pointwise norm continuous path of completely positive contractions. For n < m, set φ_{t}^{m,n} = φ_{t}^{m−1} ∘ φ_{t}^{m−2} ∘ ⋯ ∘ φ_{t}^{n} : A_{n} → A_{m}. Assume that the following holds:

For k ∈ ℕ, a, b ∈ A_{k} and ϵ > 0, there is a M ∈ ℕ such that

sup_{t∈[0,1]} ∥φ_{t}^{n,m}(φ_{t}^{m,k}(a)φ_{t}^{m,k}(b)) − φ_{t}^{n,k}(a)φ_{t}^{n,k}(b)∥ < ϵ (6)

for all M ≤ m < n.

For any C^{∗}-algebra A, set IA = C[0, 1] ⊗ A. Given the above, define φ_{m,n} : IA_{n} → IA_{m} by

φ_{m,n}(f)(t) = φ_{t}^{m,n}(f(t)).

Then (IA_{n}, φ_{n,m}) is a generalized inductive system of C^{∗}-algebras in the sense of [BK]. We consider the corresponding inductive limit C^{∗}-algebras \underrightarrow{lim}(IA_{n}, φ_{m,n}).

Lemma 5.1. 1) For each t ∈ [0, 1] there is a surjective ∗-homomorphism

π_{t} : \underrightarrow{lim}(IA_{n}, φ_{m,n}) → \underrightarrow{lim}(A_{n}, φ_{t}^{m,n}).

2) ker π_{t} = \underrightarrow{lim}(I_{t}A_{n}, φ_{m,n}), where I_{t}A_{n} = \{f ∈ IA_{n} : f(t) = 0\}.

3) For every x ∈ \underrightarrow{lim}(IA_{n}, φ_{m,n}), ∥x∥ = sup_{t∈[0,1]} ∥π_{t}(x)∥.

4) \underrightarrow{lim}(IA_{n}, φ_{m,n}) is a C[0, 1]-module such that π_{t}(fx) = f(t)π_{t}(x), f ∈ C[0, 1], x ∈ \underrightarrow{lim}(IA_{n}, φ_{m,n}).

Proof. 1) Let e_{t} : IA_{n} → A_{n} denote evaluation at t ∈ [0, 1]. Then e_{t} ∘ φ_{m,n} = φ_{t}^{m,n} ∘ e_{t} and we get a ∗-homomorphism π_{t} : \underrightarrow{lim}(IA_{n}, φ_{m,n}) → \underrightarrow{lim}(A_{n}, φ_{t}^{m,n}) by 2.3 of [BK]. π_{t} is surjective since each e_{t} is.

2) Clearly, \underrightarrow{lim}(I_{t}A_{n}, φ_{m,n}) ⊆ ker π_{t}. Let x ∈ ker π_{t}, ϵ > 0. There is a k ∈ ℕ and an element f ∈ IA_{k} such that ∥x − φ_{∞,k}(f)∥ < ϵ. Then ∥π_{t}(φ_{∞,k}(f))∥ < ϵ which implies that there is a m ≥ k such that ∥φ_{m,k}(f)(t)∥ = ∥φ_{t}^{m,k}(f(t))∥ < ϵ. There is therefore an element g ∈ I_{t}A_{m} such that ∥g − φ_{m,k}(f)∥ < ϵ. It follows that φ_{∞,m}(g) ∈ \underrightarrow{lim}(I_{t}A_{n}, φ_{m,n}) and ∥x − φ_{∞,m}(g)∥ < 2ϵ.

3) It suffices to show that π_{t}(x) = 0 ∀t ∈ [0, 1] ⇒ x = 0. So let f ∈ IA_{n} and assume that ∥π_{t}(φ_{∞,n}(f))∥ < ϵ for all t ∈ [0, 1]. For a fixed t ∈ [0, 1] there is then a k > n and an open neighbourhood U_{t} of t such that ∥φ_{s}^{k,n}(f(s))∥ < ϵ for all s ∈ U_{t}. Since φ_{s}^{m,k} is a contraction we find that ∥φ_{s}^{m,n}(f(s))∥ < ϵ for all s ∈ U_{t} and all m ≥ k. By compactness of [0, 1] we can then find a N ∈ ℕ such that ∥φ_{m,n}(f)∥ < ϵ for all m ≥ N, proving that ∥φ_{∞,n}(f)∥ < ϵ.

4) follows immediately from the observation that φ_{m,n}(fa) = fφ_{m,n}(a), a ∈ IA_{n}, f ∈ C[0, 1], m ≥ n. □

It follows from Lemma 5.1 that \underrightarrow{lim}(IA_{n}, φ_{m,n}) is a bundle of C^{∗}-algebras over [0, 1] in the sense of [KW]. The bundle is always upper semi-continuous, but to obtain a continuous bundle we need to add an additional assumption:

For n ∈ ℕ, a ∈ A_{n} and ϵ > 0, there is a m > n such that

∥φ_{t}^{m,n}(a)∥ − ϵ < ∥φ_{t}^{k,n}(a)∥ (7)

for all k ≥ m and all t ∈ [0, 1].

Lemma 5.2. Assume that (7) holds. Then (\underrightarrow{lim}(IA_{n}, φ_{m,n}), [0, 1], π) is a continuous bundle of C^{∗}-algebras.

Proof. By Lemma 5.1 it only remains to establish the continuity of t ↦ ∥π_{t}(x)∥ for an arbitrary element x ∈ \underrightarrow{lim}(IA_{n}, φ_{m,n}). Let ϵ > 0. There is a n ∈ ℕ and an element g ∈ IA_{n} such that ∥x−φ_{∞,n}(g)∥ < ϵ. Then ∥π_{t}(φ_{∞,n}(g))∥ = ∥φ_{t}^{∞,n}(g(t))∥ for all t ∈ [0, 1]. Since {∥φ_{t}^{k,n}(a)∥}_{k>n} decreases towards ∥φ_{t}^{∞,n}(a)∥ for all a ∈ A_{n} and since {g(t) : t ∈ [0, 1]} is a compact subset of A_{n}, it follows from (7) that there is a m > n such that |∥φ_{t}^{∞,n}(g(t))∥ − ∥φ_{t}^{m,n}(g(t))∥| < ϵ for all t ∈ [0, 1]. Then |∥π_{t}(x)∥ − ∥φ_{t}^{m,n}(g(t))∥| < 2ϵ for all t and t ↦ ∥φ_{t}^{m,n}(g(t))∥ is continuous, so we are done. □

Definition 5.3. Two continuous bundles of C^{∗}-algebras, (A, X, π) and (A′, Y, π′), are weakly isomorphic when there is a homeomorphism χ : Y → X and a ∗- isomorphism φ : A → A′ such that φ(fa) = f ∘ χφ(a), f ∈ C(X), a ∈ A. When X = Y and χ can be taken to be the identity map we say that the bundles are isomorphic.

Let (A, [0, 1], π) be a continuous bundle of C^{∗}-algebras. When U ⊆ [0, 1] is a relatively open subset we set

A_{U} = \overline{C_{0}(U)A} = \{x ∈ A : π_{t}(x) = 0, t ∉ U\},

which is a closed twosided ideal in A. When X = U ∩ F where U and F are relatively open and closed in [0, 1], respectively, we set A_{X} = A_{U}/A_{U∩F^{c}}. For each t ∈ X the map π_{t} : A → A_{t} induces a surjective ∗-homomorphism A_{X} → A_{t} which we again denote by π_{t}. In this way (A_{X}, X, π) becomes a continous bundle of C^{∗}-algebras over X. Up to isomorphism this construction does not depend on the way X is realized as the intersection of a closed and an open subset of [0, 1].

Definition 5.4. Let (A, [0, 1], π) be a continuous bundle of C^{∗}-algebras. A point α ∈ [0, 1] is called a point of triviality for the bundle when there is an open neighborhood U of α in [0, 1] such that (A_{U}, U, π) is a trivial bundle. A point α ∈]0, 1[ is called a point of right-sided (resp. left-sided) non-triviality when there is an ϵ > 0 such that (A_{]α−ϵ,α]},]α − ϵ, α], π) and (A_{]α,α+ϵ[},]α, α + ϵ[, π) (resp. (A_{]α−ϵ,α[},]α − ϵ, α[, π) and (A_{[α,α+ϵ[}, [α, α + ϵ[, π)) are trivial bundles. A point of non-triviality is a point which is either a right-sided or a left-sided point of nontriviality.^{3}

Definition 5.5. A continuous bundle of C^{∗}-algebras, (A, [0, 1], π), is called piecewise trivial when there is a finite set of points x_{1} < x_{2} < ⋯ < x_{k} in ]0, 1[ each of which is a point of non-triviality, while all points of [0, 1]\{x_{1}, x_{2}, …, x_{k}} are points of triviality for the bundle.

Definition 5.6. An extension of C^{∗}-algebras

\xymatrix{0 \ar[r] & J \ar[r] & E \ar[r]^{p} & A \ar[r] & 0}

is discrete asymptotically semi-split when there is a discrete completely positive asymptotic homomorphism χ_{n} : A → E, n ∈ ℕ, such that lim_{n→∞} p ∘ χ_{n}(a) = a for all a ∈ A. A bundle of C^{∗}-algebras (A, [0, 1], π) is called discretely asymptotically semi-split (resp. semi-split) when

\xymatrix{0 \ar[r] & ker π_{t} \ar[r] & A \ar[r]^{π_{t}} & A_{t} \ar[r] & 0} (8)

is discrete asymptotically semi-split (resp. semi-split) for all t ∈ [0, 1].

Note that it follows from Theorem 6 of [A] that an extension (and hence also a bundle of C^{∗}-algebras) which is discrete asymptotically semi-split is also semi-split.

Now we strengthen the assumptions on the given sequence φ_{t}^{n} in order to use Lemma 5.2 to produce continuous bundles which are piecewise trivial and discrete asymptotically semi-split, and at the same time arrange that the kernels of the fiber maps are all semi-contractible. For m > n and \underline{t} = (t_{1}, t_{2}, t_{3}, …) ∈ [0, 1]^{∞}, set

φm,n ; \underline{t} = φ_{t_{m−1}}^{m−1} ∘ φ_{t_{m−2}}^{m−2} ∘ ⋯ ∘ φ_{t_{n}}^{n} .

We can then consider the following properties of which the two first are stronger than (6) and (7), respectively.

For k ∈ ℕ, a, b ∈ A_{k} and ϵ > 0, there is a M ∈ ℕ such that

sup_{\underline{t}∈[0,1]^{∞}} ∥φn,m ; \underline{t}(φm,k ; \underline{t}(a)φm,k ; \underline{t}(b)) − φn,k ; \underline{t}(a)φn,k ; \underline{t}(b)∥ < ϵ (9)

for all M ≤ m < n.
For n ∈ ℕ, a ∈ A_{n} and ϵ > 0, there is a m > n such that

∥φm,n ; \underline{t}(a)∥ − ϵ < ∥φk,n ; \underline{t}(a)∥ (10)

for all k ≥ m and all \underline{t} ∈ [0, 1]^{∞}.

3I apologize for the fact that with this terminology an interior point of triviality is also a point of non-triviality, and that a point of non-triviality may in fact be a point of triviality. Note that we need not consider points of two-sided non-triviality.

There are completely positive contractions p_{k} : A_{k+1} → A_{k} such that

p_{k+1} ∘ φ_{s}^{k+1} ∘ φ_{t}^{k} = φ_{t}^{k} (11)

for all s, t ∈ [0, 1], s ≥ t, and all k.

Proposition 5.7. Assume that (9), (10) and (11) hold and let (A, [0, 1], π) be the continuous bundle from Lemma 5.2. There is then a piecewise trivial and discrete asymptotically semi-split continuous bundle (A′, [0, 1], π′) with only one point of non-triviality (a right-sided non-triviality) such that π_{0}(A) ≃ π′_{0}(A′), π_{1}(A) ≃ π′_{1}(A′), and such that ker π_{t} is semi-contractible for all t ∈ [0, 1].

Proof. For each n ∈ ℕ let h_{n} : [0, 1] → [0, 1] be the function

h_{n}(t) = \begin{cases}0, & t ∈ [0, \frac{1}{2}] \\ 2nt − n, & t ∈ [\frac{1}{2}, \frac{1}{2} + \frac{1}{2n}] \\ 1, & t ∈ [\frac{1}{2} + \frac{1}{2n}, 1] \end{cases}

Set ψ_{t}^{n} = φ_{h_{n}(t)}^{n}. It follows from (9) and (10) that the sequence ψ_{t}^{n} also satisfies (9) and (10), and in particular also (6) and (7). Thus (\underrightarrow{lim}(IA_{n}, ψ_{m,n}), [0, 1], π) is a continuous bundle of C^{∗}-algebras by Lemma 5.2. In order not to confuse it with the original bundle we denote it by (A′, [0, 1], π′). It follows from 2) of Lemma 5.1 that ker π′_{t} is the inductive limit of the sequence

\xymatrix{I_{t}A_{1} \ar[r]^{ψ_{1}} & I_{t}A_{2} \ar[r]^{ψ_{2}} & I_{t}A_{3} \ar[r]^{ψ_{3}} & ⋯}

By using that h_{n+1} ≥ h_{n}, it follows from (11) that the completely positive contractions p̃_{k} : I_{t}A_{k+1} → I_{t}A_{k} given by p̃_{k}(g)(s) = p_{k}(g(s)) satisfy that p̃_{k+1}∘ψ_{k+1}∘ψ_{k} = ψ_{k}. Hence the above sequence satisfies condition 2) of Definition 4.1. To see that also condition 1) of Definition 4.1 holds, observe that we can define a trivialization {ψ_{λ}} of I_{t}A_{n} of the form ψ_{λ}(f)(s) = f(H_{λ}(s)), where the H_{λ}’s are appropriately chosen functions H_{λ} : [0, 1] → [0, 1]. Therefore condition 1) of Definition 4.1 follows from (9). Consequently ker π′_{t} is approximately contractible, and hence semi-contractible by Proposition 4.2. Since h_{n}(0) = 0, h_{n}(1) = 1 for all n, it is clear that π_{0}(A) ≃ π′_{0}(A′), π_{1}(A) ≃ π′_{1}(A′). To prove triviality over [0, \frac{1}{2}] and ]\frac{1}{2}, 1] note first that \overline{C_{0}(]\frac{1}{2}, 1])A} = \underrightarrow{lim}(C_{0}(]\frac{1}{2}, 1], A_{n}), ψ^{m,n}). It follows that A_{[0, \frac{1}{2} ]} = A/A_{] \frac{1}{2} ,1]} ≃ \underrightarrow{lim}(C([0, \frac{1}{2}], A_{n}), ψ^{m,n}) and since ψ_{t}^{m,n} = φ_{0}^{m,n}, t ∈ [0, \frac{1}{2}], we find that A_{[0, \frac{1}{2} ]} ≃ C([0, \frac{1}{2}], D) where D = \underrightarrow{lim}(A_{n}, φ_{0}^{m,n}), also as C[0, \frac{1}{2}]-modules. To prove triviality over ]\frac{1}{2}, 1], consider a n ∈ ℕ and an element a ∈ C_{0}(]\frac{1}{2}, 1], A_{n}). It is then clear that there is an m ≥ n such that

sup_{t∈] \frac{1}{2} ,1]} ∥ψ_{t}^{k,m} ∘ ψ_{t}^{m,n}(a(t)) − φ_{1}^{k,m} ∘ ψ_{t}^{m,n}(a(t))∥ < ϵ

and

sup_{t∈] \frac{1}{2} ,1]} ∥ψ_{t}^{k,m} ∘ φ_{1}^{m,n}(a(t)) − φ_{1}^{k,m} ∘ φ_{1}^{m,n}(a(t))∥ < ϵ

for all k ≥ m. It follows that the identity maps on C_{0}(]\frac{1}{2}, 1], A_{n}) serves to give us an approximate intertwining in the sense of [BK], 2.3, and hence we see that A_{] \frac{1}{2} ,1]} ≃ C_{0}(]\frac{1}{2}, 1], B), also as C_{0}(]\frac{1}{2}, 1])-modules, where B = \underrightarrow{lim}(A_{n}, φ_{1}^{m,n}).

It remains to prove that the extensions (8) are all discrete asymptotically semisplit. Fix a t ∈ [0, 1]. By (11) there is a sequence of completely positive contractions p_{k} : A_{k+1} → A_{k} such that p_{k+1} ∘ ψ_{t}^{k+1} ∘ ψ_{t}^{k} = ψ_{t}^{k} for all k. As observed in the proof of Proposition 4.2 this gives us a sequence of completely positive contractions \overline{p}_{n} : A_{t} = \underrightarrow{lim}(A_{n}, ψ_{t}^{m,n}) → A_{n} such that \overline{p}_{n} ∘ ψ_{t}^{∞,l} = ψ_{t}^{n,l} when n > l. Let c_{n} : A_{n} → IA_{n} be the embedding which identifies an element of A_{n} with the corresponding constant A_{n}-valued function on [0, 1] and define χ_{n} : A_{t} → A by χ_{n}(x) = ψ_{∞,n}∘c_{n}∘\overline{p}_{n}(x). To see that \{χ_{n}\} is a discrete asymptotic homomorphism, let x, y ∈ A_{l}. For any ϵ > 0 there is then a m > l so large that ∥ψ_{t}^{∞,l}(x)ψ_{t}^{∞,l}(y) − ψ_{t}^{∞,m}(ψ_{t}^{m,l}(x)ψ_{t}^{m,l}(x))∥ < ϵ. Set x′ = ψ_{t}^{m,l}(x), y′ = ψ_{t}^{m,l}(y). For n > m we have that

∥χ_{n}(ψ_{t}^{∞,l}(x))χ_{n}(ψ_{t}^{∞,l}(y)) − χ_{n}(ψ_{t}^{∞,l}(x)ψ_{t}^{∞,l}(y))∥

≤ lim sup_{k→∞} sup_{s∈[0,1]} ∥ψ_{s}^{k,n} ∘ ψ_{t}^{n,m}(x′)ψ_{s}^{k,n} ∘ ψ_{t}^{n,m}(y′) − ψ_{s}^{k,n} ∘ ψ_{t}^{n,m}(x′y′)∥ + ϵ

≤ lim sup_{k→∞} sup_{\underline{t}∈[0,1]^{∞}} ∥φk,n ; \underline{t}(φn,m ; \underline{t}(x′))φk,n ; \underline{t}(φn,m ; \underline{t}(y′)) − φk,m ; \underline{t}(x′y′)∥ + ϵ.

It follows from (9) that the last expression is ≤ 2ϵ if just n is large enough and hence {χ_{n}} is a discrete asymptotic homomorphism. Since π_{t} ∘ χ_{n} ∘ ψ_{t}^{∞,l}(x) = ψ_{t}^{∞,n} ∘ ψ_{t}^{n,l}(x) = ψ_{t}^{∞,l}(x) for x ∈ A_{l}, l < n, we see that lim_{n→∞} π_{t} ∘ χ_{n}(z) = z for all z ∈ A_{t}. This shows that (8) is discrete asymptotically semi-split. □

Observe that if all the A_{n}’s are separable and/or nuclear C^{∗}-algebras it follows that the continuous bundles obtained here, in Lemma 5.2 and in Proposition 5.7, are separable and/or nuclear (in the sense that the bundle C^{∗}-algebra is separable and/or nuclear.) For the nuclearity part of this assertion, use Proposition 5.1.3 of [BK].

6. From KK-equivalence to continuous bundles

In this section we consider two separable C^{∗}-algebras A and B. For simplicitiy of notation we assume first that both are stable. Recall from Theorem 4.1 of [Th2] that there is a completely positive asymptotic homomorphism ν^{A} : cone(A) → SA with the property that when ψ, φ : SA → SA are completely positive asymptotic homomorphisms such that [φ] = [ψ] in [[SA, SA]]_{cp}, then there is a norm continuous path U_{t}, t ∈ [1, ∞), of unitaries in M_{2}(SA)^{+} and an increasing continuous function r : [1, ∞) → [1, ∞) such that

lim_{t→∞} U_{t} ( \begin{matrix}ψ_{t}(a) \\ & ν_{r(t)}^{A}(a) \end{matrix} ) U_{t}^{∗} − ( \begin{matrix}φ_{t}(a) \\ & ν_{r(t)}^{A}(a) \end{matrix} ) = 0

for all a ∈ SA.

Lemma 6.1. 1) lim_{t→∞} ∥ν_{t}^{A}(a)∥ = ∥a∥ for all cone(A).

2) Every element of [[SA, B]]_{cp} is represented by a completely positive asymptotic homomorphism φ = (φ_{t})_{t∈[1,∞)} : SA → B with the property that lim_{t→∞} ∥φ_{t}(a)∥ = ∥a∥ for all a ∈ SA.

Proof. We prove 1) and 2) in one stroke. Let λ : cone(A) → B be the completely positive asymptotic homomorphism which features in Theorem 4.1 of [Th2]. Since [φ ⊕ λ] = [φ] in [[SA, B]]_{cp} it suffices to show that lim_{t→∞} ∥λ_{t}(a)∥ = ∥a∥ for all a ∈ cone(A). (With B = SA this will prove 1).) λ has the form λ_{t}(a) = p_{t}π(a)p_{t}, where π : cone(A) → M(B) is an absorbing ∗-homomorphism and (p_{t})_{t∈[1,∞)} is a norm continuous path of positive elements in M(B) with properties described in Theorem 3.7 of [Th2]. Since π is absorbing and there is an injective ∗-homomorphism cone(A) → M(B) (because B is stable) it follows that π is injective. Let a ∈ cone(A) and ϵ > 0. There is then an element b ∈ B, ∥b∥ ≤ 1, such that ∥a∥ − ϵ < ∥π(a)b∥. Since lim_{t→∞} p_{t}b = b and lim_{t→∞} p_{t}π(a) − π(a)p_{t} = 0 it follows that ∥λ_{t}(a)∥ ≥ ∥p_{t}π(a)p_{t}b∥ > ∥a∥ − ϵ for all large t. □

Remark 6.2. It follows from Lemma 6.1 and [L] that for any pair of separable C^{∗}-algebras it holds that the stable suspension of any one of them is a deformation of the stable suspension of the other.

A map μ : D → C is said to be δ-multiplicative on a subset F ⊆ D when ∥μ(a)μ(b) − μ(ab)∥ < δ for all a, b ∈ F.

Lemma 6.3. Let φ : SA → SB and ψ : SB → SA be completely positive asymptotic homomorphisms such that [ψ] • [φ] = [id_{SA}] in [[SA, SA]]_{cp}. It follows that there is a continuous function r : [1, ∞) → [1, ∞) with the following property: When F ⊆ SA and G ⊆ cone(A) are compact subsets, δ > 0 and t_{0} ∈ [1, ∞), there is a t_{1} ≥ t_{0}, and for each s ≥ r(t_{1}) a unitary T_{s} ∈ M_{2}(SA)^{+} and a completely positive contraction μ_{s} : cone(A) → SA such that μ_{s} is δ-multiplicative on G, ∥μ_{s}(a)∥ ≥ ∥a∥ − δ, a ∈ G, and

∥T_{s} (\begin{matrix}ψ_{s} ∘ φ_{t_{1}}(a) \\ & μ_{s}(a) \end{matrix}) T_{s}^{∗} − (\begin{matrix}a \\ & μ_{s}(a) \end{matrix}) ∥ < δ

for all a ∈ F.

Proof. By definition of the composition product there is a continuous function r : [1, ∞) → [1, ∞) such that [ψ] • [φ] is represented by {ψ_{r′(t)} ∘ φ_{t}}_{t∈[1,∞)} for any continuous function r′ : [1, ∞) → [1, ∞) such that r′ ≥ r. We may assume that r is convex. We claim that r then has the stated property, and we prove it by contradiction. So assume F, G, δ and t_{0} ∈ [1, ∞) is a quadruple for which the stated property fails. There is then a sequence t_{0} < t_{1} < t_{2} < t_{3} < … in [1, ∞) such that lim_{i→∞} t_{i} = ∞ and for each i a s_{i} ≥ r(t_{i}) with the property that

sup_{a∈F} ∥T (\begin{matrix}ψ_{s_{i}} ∘ φ_{t_{i}}(a) \\ & μ(a) \end{matrix}) T^{∗} − (\begin{matrix}a \\ & μ(a) \end{matrix}) ∥ ≥ δ (12)

for every unitary T ∈ M_{2}(SA)^{+} and every completely positive contraction μ : cone(A) → SA which is δ-multiplicative on G and satisfies that ∥μ(a)∥ ≥ ∥a∥ − δ, a ∈ G. For s ∈ [t_{i}, t_{i+1}], write s = αt_{i+1} + (1 − α)t_{i}, where α ∈ [0, 1], and set λ_{s} = ψ_{αs_{i+1}+(1−α)s_{i}} ∘ φ_{αt_{i+1}+(1−α)t_{i}}. Since αs_{i+1} + (1 − α)s_{i} ≥ αr(t_{i+1}) + (1 − α)r(t_{i}) ≥ r(αt_{i+1} + (1 − α)t_{i}) it follows that λ = (λ_{s})_{s∈[t_{1},∞)} is a completely positive asymptotic homomorphism. In fact λ_{t} = ψ_{v(t)} ∘ φ_{t}, t ≥ t_{1}, where v : [t_{1}, ∞) → [s_{1}, ∞) is a continuous function such that v(t) ≥ r(t), t ≥ t_{1}, and hence [λ] = [ψ]•[φ] in [[SA, SA]]_{cp}. By Theorem 4.1 of [Th2] there is a completely positive asymptotic homomorphism ν = (ν_{t})_{t∈[1,∞)} : cone(A) → SA and a path of unitaries (T_{t}) ∈ M_{2}(SA)^{+} such that

lim_{t→∞} T_{t} (\begin{matrix}λ_{t}(a) \\ & ν_{t}(a) \end{matrix}) T_{t}^{∗} − (\begin{matrix}a \\ & ν_{t}(a) \end{matrix}) = 0

for all a ∈ SA. By 1) of Lemma 6.1, ν can be chosen such that lim_{t→∞} ∥ν_{t}(a)∥ = ∥a∥ for all a ∈ cone(A). In particular, it follows that for i large enough, ν_{t_{i}} is δmultiplicative on G, ∥ν_{t_{i}}(a)∥ ≥ ∥a∥ − δ, a ∈ G, and

sup_{a∈F} ∥T_{t_{i}} (\begin{matrix}ψ_{s_{i}} ∘ φ_{t_{i}}(a) \\ & ν_{t_{i}}(a) \end{matrix}) T_{t_{i}}^{∗} − (\begin{matrix}a \\ & ν_{t_{i}}(a) \end{matrix}) ∥ < δ,

contradicting (12). □

Now assume that A and B are KK-equivalent. This means that there are completely positive asymptotic homomorphisms φ = {φ_{t}} : SA → SB and ψ = {ψ_{t}} : SB → SA such that [φ] • [ψ] = [id_{SB}] in [[SB, SB]]_{cp} and [ψ] • [φ] = [id_{SA}] in [[SA, SA]]_{cp}. By 2) of Lemma 6.1 we may assume that lim_{t→∞} ∥φ_{t}(a)∥ = ∥a∥, a ∈ SA, and lim_{t→∞} ∥ψ_{t}(b)∥ = ∥b∥, b ∈ SB. Choose a continuous function r : [1, ∞) → [1, ∞) having the property described in Lemma 6.3 and a continuous function s : [1, ∞) → [1, ∞) with the same property relative to (ψ, φ) instead of (φ, ψ). To proceed we need to introduce some notation and terminology. Set D = A ⊕ B. We will consider SA, SB, cone(A) and cone(B) as C^{∗}- subalgebras of cone(D). Observe that cone(A), cone(B), SA, SB and cone(D) are stable since A and B are. When E is any of these stable C^{∗}-algebras we choose isometries, V_{1}, V_{2}, in M(E) such that V_{1}V_{1}^{∗} + V_{2}V_{2}^{∗} = 1. For e_{1}, e_{2} ∈ E we can then define e_{1} ⊕ e_{2} = V_{1}e_{1}V_{1}^{∗} + V_{2}e_{2}V_{2}^{∗} ∈ E. A completely positive contraction SA ⊕ cone(D) → SB ⊕ cone(D) is said to be of φ-type when it has the form (a, d) ↦ (V (φ_{t}(a) ⊕ μ(d))V^{∗}, a ⊕ d) for some t ∈ [1, ∞), some unitary V ∈ (SB)^{+} and some completely positive contraction μ : cone(D) → SB. Similarly, we define a completely positive contraction SB ⊕ cone(D) → SA ⊕ cone(D) to be of ψ-type when it has the form (b, d) ↦ (W (ψ_{s}(b) ⊕ ν(d))W^{∗}, b ⊕ d) for some s ∈ [1, ∞), some unitary W ∈ (SA)^{+} and some completely positive contraction ν : cone(D) → SA. A map μ : SA ⊕ cone(D) → SA ⊕ cone(D) is called an almost identity map when it has the form μ(a, d) = (a ⊕ ν(a, d), χ(a, d) ⊕ d) where χ : SA ⊕ cone(D) → cone(D) and ν : cone(A) ⊕ cone(D) → SA are completely positive contractions. For each s ∈ [0, 1], let h_{s} : [0, 1] → [0, 1] be the continuous function such that h_{s}(t) = t, t ∈ [0, s], and h_{s}(t) = s, t ∈ [s, 1]. Define ψ_{s}^{D} : cone(D) → cone(D) by ψ_{s}^{D}(f)(t) = f(h_{s}(t)). Note that with a natural choice of isometries V_{1}, V_{2} to define ⊕ we have that ψ_{t}^{D}(a ⊕ d) = ψ_{t}^{D}(a) ⊕ ψ_{t}^{D}(d). Given an almost identity map μ as above we can use {ψ_{s}^{D}}_{s∈[0,1]} and {ψ_{s}^{A}}_{s∈[0,1]} to define maps μ^{(s)}, s ∈ [0, 1], by

μ^{(s)}(a, d) = (a ⊕ ν(ψ_{s}^{A}(a), ψ_{s}^{D}(d)), ψ_{s}^{D}(χ(a, ψ_{s}^{D}(d))) ⊕ ψ_{s}^{D}(d)) .

Given C^{∗}-algebras X, Y and Z and compact subsets F ⊆ X, G ⊆ Y, H ⊆ Z, a diagram of the form

[figure: (X, F); φ; ψ; (Y, G); λ; (Z, H); ϵ]

will mean that ϵ > 0, that φ, ψ and λ are completely positive contractions such that ψ and φ are ϵ-multiplicative on F, λ is ϵ-multiplicative on G, φ(F) ∪ λ(G) ⊆ H, ψ(F) ⊆ G, and ∥φ(x) − λ ∘ ψ(x)∥ < ϵ, x ∈ F.

Let q_{1} : SA⊕cone(D) → SA, q_{1} : SB ⊕cone(D) → SB be the projections to the first coordinate and q_{2} : SA ⊕ cone(D) → cone(D), q_{2} : SB ⊕ cone(D) → cone(D) the projections to the second.

Lemma 6.4. Let ϵ_{1} > ϵ_{2} > ϵ_{3} > … be a sequence in ]0, 1[ and let F_{1}^{0} ⊆ F_{2}^{0} ⊆ F_{3}^{0} ⊆ ⋯, G_{1}^{0} ⊆ G_{2}^{0} ⊆ G_{3}^{0} ⊆ ⋯ be compact subsets of SA ⊕ cone(D) and SB ⊕ cone(D), respectively. There is then an infinite diagram

[figure: (SA ⊕ cone(D), F_{1}); ϵ_{1}; μ_{1}; Φ_{1}; (SB ⊕ cone(D), G_{1}); ν_{1}; Ψ_{1}; (SA ⊕ cone(D), F_{2}); ϵ_{2}; Φ_{2}; μ_{2}; (SB ⊕ cone(D), G_{2}); ϵ_{1}; ν_{2}; Ψ_{2}; (SA ⊕ cone(D), F_{3}); ϵ_{3}; μ_{3}; Φ_{3}; (SB ⊕ cone(D), G_{3}); ν_{3}; Ψ_{3}; ϵ_{2}; ⋮; ⋮; ϵ_{3}]

where the Φ_{i}’s are of φ-type, the Ψ_{i}’s of ψ-type, the μ_{i}’s and ν_{i}’s are almost identity maps and

F_{n}^{0} ∪ ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} μ_{n−1}^{(t_{1})} ∘ μ_{n−2}^{(t_{2})} ∘ ⋯ ∘ μ_{j}^{(t_{n−j})}(F_{n}^{0}) ⊆ F_{n},

G_{n}^{0} ∪ ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} ν_{n−1}^{(t_{1})} ∘ ν_{n−2}^{(t_{2})} ∘ ⋯ ∘ ν_{j}^{(t_{n−j})}(G_{n}^{0}) ⊆ G_{n}

for all n. Furthermore, we arrange that μ_{n}^{s} and ν_{n}^{s} are ϵ_{n}-multiplicative on F_{n} and G_{n}, respectively, and

∥q_{1} ∘ μ_{n}^{(s)}(x)∥ ≥ ∥μ_{n}^{(s)}(x)∥ − ϵ_{n}, x ∈ F_{n},
∥q_{1} ∘ ν_{n}^{(s)}(x)∥ ≥ ∥ν_{n}^{(s)}(x)∥ − ϵ_{n}, x ∈ G_{n},

for all n and all s ∈ [0, 1].

Proof. To simply notation, set

G′_{n} = G_{n}^{0} ∪ ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} ν_{n−1}^{(t_{1})} ∘ ν_{n−2}^{(t_{2})} ∘ ⋯ ∘ ν_{j}^{(t_{n−j})}(G_{n}^{0})

and

F′_{n} = F_{n}^{0} ∪ ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} μ_{n−1}^{(t_{1})} ∘ μ_{n−2}^{(t_{2})} ∘ ⋯ ∘ μ_{j}^{(t_{n−j})}(F_{n}^{0})

We proceed by induction. So assume that we have constructed everything up to n, i.e., that we have a finite diagram as above ending with

\xymatrix{(SA ⊕ cone(D), F_{n}) \ar[r]^{Φ_{n}} & (SB ⊕ cone(D), G_{n})}

Here Φ_{n} is of φ-type, i.e., is given by (a, d) ↦ (V (φ_{k}(a)⊕μ(d))V^{∗}, a⊕d) for some k ∈ [1, ∞), some unitary V ∈ (SB)^{+} and some completely positive contraction μ : cone(D) → SB. In addition we shall assume that

A) μ is \frac{ϵ_{n}}{2} -multiplicative on ψ_{t}^{D}(q_{2}(F_{n})), t ∈ [0, 1], B) for all y ≥ r(k) there is a unitary T_{y} ∈ (SA)^{+} and a completely positive

contraction δ : cone(D) → SA which is \frac{ϵ_{n}}{2} -multiplicative on \{ψ_{t}^{D}(a ⊕ d) : (a, d) ∈ F_{n}} ∪ q_{2}(F_{n}) ∪ ψ_{t}^{D}(q_{2}(G_{n})), t ∈ [0, 1], and satisfies that

∥δ(ψ_{t}^{D}(a ⊕ d))∥ ≥ ∥ψ_{t}^{D}(d)∥ − ϵ_{n}, (a, d) ∈ F_{n}, t ∈ [0, 1], and

∥T_{y}(ψ_{y} ∘ φ_{k}(a) ⊕ δ(a ⊕ d))T_{y}^{∗} − a ⊕ δ(a ⊕ d)∥ < \frac{ϵ_{n}}{2} , (a, d) ∈ F_{n}.

This is allright if we make sure that Φ_{n+1} satisfies the corresponding n + 1-conditions, and we will. Choose y ≥ r(k) so large that there is a unitary W ∈ (SA)^{+} such that

∥W ψ_{y}(φ_{k}(a)) ⊕ ψ_{y}(μ(d)))W^{∗} − ψ_{y}(V (φ_{k}(a) ⊕ μ(d))V^{∗}))∥ < \frac{ϵ_{n}}{2} (13)

for all (a, d) ∈ F_{n}. This is possible because V ∈ (SB)^{+} and ψ is an asymptotic homomorphism. By B) there is a unitary U ∈ (SA)^{+} and a completely positive contraction ν : cone(D) → SA which is \frac{ϵ_{n}}{2} -multiplicative on \{ψ_{t}^{D}(a ⊕ d) : (a, d) ∈ F_{n}}∪q_{2}(F_{n})∪ψ_{t}^{D}(q_{2}(G_{n})), t ∈ [0, 1], and satisfies that ∥ν(ψ_{t}^{D}(a⊕d))∥ ≥ ∥ψ_{t}^{D}(d)∥− ϵ_{n}, (a, d) ∈ F_{n}, t ∈ [0, 1], and ∥U(ψ_{y} ∘ φ_{k}(a) ⊕ ν(a ⊕ d))U^{∗} − a ⊕ ν(a ⊕ d)∥ < \frac{ϵ_{n}}{2} for all (a, d) ∈ F_{n}. Set

X = ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} μ_{n−1}^{(t_{1})} ∘ μ_{n−2}^{(t_{2})} ∘ ⋯ ∘ μ_{j}^{(t_{n−j})}(F_{n+1}^{0}) , and
X′ = {q_{1} ∘ Φ_{n}(a, ψ_{t}^{D}(d)) ⊕ a ⊕ ψ_{t}^{D}(d) : (a, d) ∈ X, t ∈ [0, 1]}.

Then X and X′ are both compact subsets of SA ⊕ cone(D) and cone(D), respectively. By Lemma 6.3 we can arrange, by increasing y further, that

C) For all z ≥ s(y) there is a unitary T_{z} ∈ (SB)^{+} and a completely positive contraction δ′ : cone(D) → SB which is \frac{ϵ_{n+1}}{2} -multiplicative on ψ_{t}^{D}(\{b ⊕ d : (b, d) ∈ G_{n}} ∪ {q_{1} ∘ Φ_{n}(a, d) ⊕ a ⊕ d : (a, d) ∈ F_{n}} ∪ q_{2}(G_{n}) ∪ X′ ∪ q_{2}(F_{n+1}^{0})), t ∈ [0, 1], and satisfies that ∥δ′(ψ_{t}^{D}(b⊕d))∥ ≥ ∥ψ_{t}^{D}(d)∥−ϵ_{n+1}, for (b, d) ∈ G_{n}, t ∈ [0, 1], and

∥T_{z}(φ_{z} ∘ ψ_{y}(b) ⊕ δ′(b ⊕ d))T_{z}^{∗} − b ⊕ δ′(b ⊕ d)∥ < \frac{ϵ_{n}}{2}

for all (b, d) ∈ G_{n}.

In addition we can arrange that ψ_{y} is \frac{ϵ_{n}}{2} -multiplicative on q_{1}(G_{n})∪μ(ψ_{t}^{D}(q_{2}(F_{n}))), t ∈ [0, 1]. Observe that there is a unitary T ∈ (SA)^{+} such that

∥T (ψ_{y} ∘ φ_{k}(a) ⊕ ψ_{y} ∘ μ(d) ⊕ ν(a ⊕ d))T^{∗} − a ⊕ ψ_{y} ∘ μ(d) ⊕ ν(a ⊕ d)∥ < \frac{ϵ_{n}}{2} (14)

for all (a, d) ∈ F_{n}. Define Ψ_{n} : SB ⊕ cone(D) → SA ⊕ cone(D) by

Ψ_{n}(b, d) = (T (W^{∗}ψ_{y}(b)W ⊕ ν(d))T^{∗}, b ⊕ d) ,

and μ_{n} : SA ⊕ cone(D) → SA ⊕ cone(D) by

μ_{n}(a, d) = (a ⊕ ψ_{y} ∘ μ(d) ⊕ ν(a ⊕ d), V (φ_{k}(a) ⊕ μ(d))V^{∗} ⊕ a ⊕ d) .

Then Ψ_{n} is of ψ-type and μ_{n} is an almost identity map. It is straightforward to see that Ψ_{n} and μ_{n} are ϵ_{n}-multiplicative on G_{n} and F_{n}, respectively, so (14) and (13) give us the diagram

[figure: (SA ⊕ cone(B), F_{n}); μ_{n}; Φ_{n}; (SB ⊕ cone(A), G_{n}); Ψ_{n}; (SA ⊕ cone(B), F_{n+1}); ϵ_{n}]

where F_{n+1} = Ψ_{n}(G_{n}) ∪ μ_{n}(F_{n}) ∪ ⋃_{t∈[0,1]} μ_{n}^{(t)}(X) ∪ F_{n+1}^{0}. To check that the ad-

ditional requirements on μ_{n}, concerning μ_{n}^{(s)}, s ∈ [0, 1], are also satisfied, observe first of all that F′_{n+1} ⊆ F_{n+1}. Combine the fact that ψ_{y} is \frac{ϵ_{n}}{2} -multipliciative on μ(ψ_{t}^{B}(q_{2}(F_{n}))) with A) to see that (a, d) ↦ ψ_{y} ∘μ(ψ_{t}^{D}(d)) is ϵ_{n}-multiplicative on F_{n} for all t. It follows from A) that (a, d) ↦ V (φ_{k}(a)⊕μ(ψ_{t}^{D}(d)))V^{∗} is ϵ_{n}-multiplicative on F_{n} for all t, and one of the requirements on ν was that (a, d) ↦ ν(ψ_{t}^{D}(a ⊕ d)) is ϵ_{n}-multiplicative on F_{n} for all t. By putting all of this together we see that μ_{n}^{(s)} is ϵ_{n}-multiplicative on F_{n} for all s ∈ [0, 1]. Another requirement on ν was that ∥ν(ψ_{t}^{D}(a ⊕ d))∥ ≥ ∥ψ_{t}^{D}(d)∥ − ϵ_{n} for all t and all (a, d) ∈ F_{n}. It follows that ∥q_{1} ∘ μ_{n}^{(s)}(x)∥ ≥ ∥μ_{n}^{(s)}(x)∥ − ϵ_{n} for all x ∈ F_{n} and all s ∈ [0, 1]. We claim that

A′) ν is ϵ_{n}-multiplicative on ψ_{t}^{D}(q_{2}(G_{n})), t ∈ [0, 1], B′) for all z ≥ s(y) there is a unitary T_{z} ∈ (SB)^{+} and a completely positive

contraction δ′ : cone(D) → SB which is \frac{ϵ_{n+1}}{2} -multiplicative on \{ψ_{t}^{D}(b ⊕ d) : (b, d) ∈ G_{n}} ∪ q_{2}(G_{n}) ∪ ψ_{t}^{D}(q_{2}(F_{n+1})), t ∈ [0, 1], and satisfies that

∥δ′(ψ_{t}^{D}(b ⊕ d))∥ ≥ ∥ψ_{t}^{D}(d)∥ − ϵ_{n+1}, (b, d) ∈ G_{n}, and

∥T_{z}(φ_{z} ∘ ψ_{y}(b) ⊕ δ′(b ⊕ d))T_{z}^{∗} − b ⊕ δ′(b ⊕ d)∥ < \frac{ϵ_{n+1}}{2} , (b, d) ∈ G_{n}.

A′) was one of the requirements on ν. To see that B′) holds, note that q_{2}(Ψ_{n}(G_{n})) = {b ⊕ d : (b, d) ∈ G_{n}}, that q_{2}(μ_{n}(F_{n})) = {q_{1} ∘ Φ_{n}(a, d) ⊕ a ⊕ d : (a, d) ∈ F_{n}} and that q_{2}(μ_{n}^{(t)}(X)) ⊆ ψ_{t}^{D}(X′). Therefore B′) follows from C).

Now we can exchange the role of φ and ψ and construct in the same way a diagram

[figure: ϵ_{n}; (SB ⊕ cone(D), G_{n}); ν_{n}; Ψ_{n}; (SA ⊕ cone(D), F_{n+1}); Φ_{n+1}; (SB ⊕ cone(D), G_{n+1})]

where G′_{n+1} ⊆ G_{n+1}. Furthermore by using Lemma 6.3 as it was used to obtain C) above we can arrange that Φ_{n+1} satisfies the n + 1-version of B). The n + 1-version of A) follows from the constructions by use of B′). In this way we obtain the desired diagram by induction. □

Let F_{1}^{0} ⊆ F_{2}^{0} ⊆ F_{3}^{0} ⊆ ⋯ and G_{1}^{0} ⊆ G_{2}^{0} ⊆ G_{3}^{0} ⊆ ⋯ be sequences of finite sets with dense union in SA ⊕ cone(D) and SB ⊕ cone(D), respectively. By combining Lemma 6.4 with 2.4 of [BK] we get sequences of almost identity maps, φ_{n} : SA ⊕ cone(D) → SA ⊕ cone(D) and ψ_{n} : SB ⊕ cone(D) → SB ⊕ cone(D), such that \underrightarrow{lim}(SA ⊕ cone(D), φ_{m,n}) ≃ \underrightarrow{lim}(SB ⊕ cone(D), ψ_{m,n}) and

I) φ_{n}^{(t)} is 2^{−n}-multiplicative on F_{n}, t ∈ [0, 1], II) ∥q_{1} ∘ φ_{n}^{(t)}(x)∥ ≥ ∥φ_{n}^{(t)}(x)∥ − 2^{−n}, x ∈ F_{n}, t ∈ [0, 1],

where

F_{n} = F_{n}^{0} ∪ ⋃_{j<n} ⋃_{(t_{1},...,t_{n−j})∈[0,1]^{n−j}} φ_{n−1}^{(t_{1})} ∘ φ_{n−2}^{(t_{2})} ∘ ⋯ ∘ φ_{j}^{(t_{n−j})}(F_{n}^{0}).

We arrange also that (ψ_{n}, G_{n}) satisfy the analogues of I) and II). In order to make connection with the last section, set φ_{t}^{n} = φ_{n}^{(t)}. We check that (9) holds : Let a, b ∈ SA ⊕ cone(D), k ∈ ℕ and ϵ > 0 be given. It follows from the density of ⋃_{m} F_{m}^{0} in SA ⊕ cone(D) that there is a M ∈ ℕ and elements x, y ∈ F_{M}^{0} such that

|∥φ(λ(x)λ(y)) − φ(λ(x))φ(λ(y))∥ − ∥φ(λ(a)λ(b)) − φ(λ(a))φ(λ(b))∥| < \frac{ϵ}{2} (15)

for all linear contractions φ, λ : SA ⊕ cone(D) → SA ⊕ cone(D). By increasing M we may assume M > k and that 2^{−M+1} < \frac{ϵ}{2}. It follows from I) that

∥φn,m ; \underline{t}(φm,k ; \underline{t}(x)φm,k ; \underline{t}(y)) − φn,m ; \underline{t}(φm,k ; \underline{t}(x))φn,m ; \underline{t}(φm,k ; \underline{t}(y))∥ ≤ ∑_{j=m}^{n} 2^{−j} < \frac{ϵ}{2} (16)

for all n > m ≥ M and all \underline{t} ∈ [0, 1]^{∞}. By combining (15) and (16) we get (9).

To prove that (10) holds observe that by the nature of an almost identity map

and II) we have that ∥φn+1,n ; \underline{t}(x)∥ ≥ ∥φm,n ; \underline{t}(x)∥ ≥ ∥φn+1,n ; \underline{t}(x)∥ − 2^{−n} for all x ∈ F_{n},

all \underline{t} ∈ [0, 1]^{∞} and all m > n. So if a ∈ SA ⊕ cone(D), n ∈ ℕ and ϵ > 0 are given we choose a m > n and an element x ∈ F_{m}^{0} such that 2^{−m} < \frac{ϵ}{3} and ∥a − x∥ < \frac{ϵ}{3}. It follows then that

∥φk,n ; \underline{t}(a)∥ ≥ ∥φk,n ; \underline{t}(x)∥ − \frac{ϵ}{3} ≥ ∥φm+1,n ; \underline{t}(x)∥ − \frac{2ϵ}{3} > ∥φm+1,n ; \underline{t}(a)∥ − ϵ

for all \underline{t} ∈ [0, 1]^{∞} and all k > m. Finally, we must consider (11). It is apparent from the definition of an almost identity map that there is a completely positive contraction p : SA⊕cone(D) → SA⊕cone(D) with the property that p∘φ_{t}^{n}(a, d) = (a, ψ_{t}^{D}(d)), t ∈ [0, 1], (a, d) ∈ SA ⊕ cone(D). Since ψ_{s}^{D} ∘ ψ_{t}^{D} = ψ_{t}^{D}, s ≥ t, it follows from the form of φ_{t}^{n}, that p ∘ φ_{s}^{n+1} ∘ φ_{t}^{n} = φ_{t}^{n} for s ≥ t. We can therefore use p as p_{k} for all k in (11).

Having established both (9), (10) and (11), Proposition 5.7 gives us a continuous bundle of C^{∗}-algebras, (A, [0, 1], π), which is discrete asymptotically semisplit and piecewise trivial with only one point of non-triviality such that ker π_{t} is semi-contractible for all t ∈ [0, 1] and such that the fibers at 0 and 1 are \underrightarrow{lim}(SA ⊕ cone(D), φ_{m,n}^{(0)}) and \underrightarrow{lim}(SA ⊕ cone(D), φ_{m,n}), respectively. Note that

\underrightarrow{lim}(SA⊕cone(D), φ_{n,m}^{(0)}) ≃ SA since SA is stable and φ_{n}^{(0)} has the form φ_{n}^{(0)}(a, d) = (V aV^{∗}, 0) for the same isometry V ∈ M(SA). Therefore, when we apply the same procedure to SB ⊕ cone(D), we get all together the following result.

Theorem 6.5. Let A and B be KK-equivalent separable C^{∗}-algebras. It follows that there are separable continuous bundles, (A, [0, 1], π) and (A′, [0, 1], π′), of C^{∗}- algebras, which are discrete asymptotically semi-split and piecewise trivial with only one point of non-triviality, such that ker π_{t} and ker π′_{t} are semi-contractible for all t ∈ [0, 1], π_{0}(A) ≃ SA ⊗ K, π′_{0}(A′) ≃ SB ⊗ K and π_{1}(A) ≃ π′_{1}(A′).

7. Concatenation of bundles

Definition 7.1. Let A be a C^{∗}-algebra. A finite semi-split decomposition series for A consists of a series of ideals

A = I_{n} ⊇ I_{n−1} ⊇ I_{n−2} ⊇ ⋯ ⊇ I_{1} ⊇ I_{0} = {0}

such that the corresponding extensions 0 → I_{k−1} → I_{k} → I_{k}/I_{k−1} → 0 are semisplit for k = 2, 3, …, n. The length of the decomposition series is said to be n, and the C^{∗}-algebras I_{i}/I_{i−1}, i = 1, 2, …, n, will be called the succesive quotients of the decomposition series. When n = 2 we say that A is a semi-split extension of I_{2}/I_{1} by I_{1}.

The proof of the following is left to the reader.

Lemma 7.2. Let

\xymatrix{0 \ar[r] & J \ar[r] & E \ar[r]^{p} & A \ar[r] & 0}

be a semi-split extension and J = J_{n} ⊇ J_{n−1} ⊇ J_{n−2} ⊇ ⋯ ⊇ J_{1} ⊇ J_{0} = {0}, A = A_{m} ⊇ A_{m−1} ⊇ A_{m−2} ⊇ ⋯ ⊇ A_{1} ⊇ A_{0} = {0} finite semi-split decomposition series for J and A, respectively. Set J_{n+i} = p^{−1}(A_{i}), i = 1, 2, …, m. Then

E = J_{m+n} ⊇ J_{m+n−1} ⊇ ⋯ ⊇ J_{1} ⊇ J_{0} = {0}

is a semi-split decomposition series for E such that J_{k}/J_{k−1} ≃ A_{k−n}/A_{k−n−1} for k > n.

Lemma 7.3. Let (A, [0, 1], π) and (A′, [0, 1], π′) be piecewise trivial and semi-split continuous bundles such that A_{1} ≃ A′_{0}. There is then a piecewise trivial and semisplit continuous bundle (B, [0, 1], π′′) with the following properties.

1) (B_{[0, \frac{1}{2} ]}, [0, \frac{1}{2}], π′′) and (B_{[ \frac{1}{2} ,1]}, [\frac{1}{2}, 1], π′′) are weakly isomorphic to (A, [0, 1], π) and (A′, [0, 1], π′), respectively, 2) for t ∈ [0, \frac{1}{2}] there an s ∈ [0, 1] and a semi-split extension

\xymatrix{0 \ar[r] & ker π′_{0} \ar[r] & ker π′′_{t} \ar[r] & ker π_{s} \ar[r] & 0,}

and for t ∈ [\frac{1}{2}, 1] there is an s ∈ [0, 1] and a semi-split extension

\xymatrix{0 \ar[r] & ker π_{1} \ar[r] & ker π′′_{t} \ar[r] & ker π′_{s} \ar[r] & 0.}

Proof. Let α : A_{1} → A′_{0} be a ∗-isomorphism. The C^{∗}-algebra B = {(a_{1}, a_{2}) ∈ A ⊕ A′ : α ∘ π_{1}(a_{1}) = π′_{0}(a_{2})} is the bundle C^{∗}-algebra for a bundle with the described properties. In particular, the part about the extensions being semi-split follows from the triviality of the bundles in a neighbourhood of the endpoints. □

Theorem 7.4. Let A and B be KK-equivalent separable C^{∗}-algebras. It follows that there is a separable continuous bundle of C^{∗}-algebras, (A, [0, 1], π), which is semi-split and piecewise trivial with no more than 2 points of non-triviality such that π_{0}(A) ≃ SA ⊗ K, π_{1}(A) ≃ SB ⊗ K, and ker π_{t} is a semi-split extension of semi-contractible C^{∗}-algebras for all t ∈ [0, 1].

Proof. Concatenate the two bundles from Theorem 6.5 by using Lemma 7.3 and apply Lemma 7.2. □

Definition 7.5. A separable C^{∗}-algebra D is called crossed-contractible when there is a C^{∗}-dynamical system (A, ℝ, α) with A contractible such that D ≃ A ×_{α} ℝ.

It follows from [FS] that a crossed-contractible C^{∗}-algebra is KK-contractible. It is easy to see that a crossed-contractible C^{∗}-algebra need not be locally contractible.

Theorem 7.6 (Rieffel [R], Elliott, Natsume, Nest [ENN]). Let A be a separable C^{∗}-algebra. There is then a separable semi-split and piecewise trivial continuous bundle of C^{∗}-algebras, (A, [0, 1], π), with no more than one point of non-triviality such that π_{0}(A) ≃ A ⊗ K, π_{1}(A) ≃ S^{2}A, and ker π_{t} is crossed-contractible for all t ∈ [0, 1].

Proof. For λ ∈ [0, 1], define an action α^{λ} : ℝ → Aut C_{0}(ℝ) ⊗ A by α_{s}^{λ}(f)(t) = f(t − (1 − λ)s). For λ ∈ [1, 2], set α_{s}^{λ} = id for all s. Define an action α : ℝ → Aut[C[0, 2] ⊗ C_{0}(ℝ) ⊗ A] by α_{s}(g)(λ) = α_{s}^{λ}(g(λ)), λ ∈ [0, 2]. It follows from [R] that [C[0, 2]⊗C_{0}(ℝ)⊗A]×_{α} ℝ is the bundle C^{∗}-algebra for a bundle over [0, 2] such that ker π_{t} is crossed-contractible for all t ∈ [0, 2]. It is well-known that the fiber over 0 is A ⊗ K and the fiber over 2 is S^{2}A. As pointed out in [ENN] the bundle is trivial over [0, 1[ and it is clearly trivial over [1, 2]. The fact that the bundle is semi-split follows from the Choi-Effros lifting theorem, [CE], by observing that the bundle in the general case is obtained from the nuclear bundle which results from the special case where A = ℂ by tensoring with A. □

By concatenation the bundle from Theorem 7.6 with the bundles from Theorem 7.4, applied to SA and SB, we get the following result.

Theorem 7.7. Let A and B be separable and stable KK-equivalent C^{∗}-algebras. There is then a separable continuous bundle of C^{∗}-algebras, (A, [0, 1], π), which is semi-split and piecewise trivial with no more than four points of non-triviality, and for each t ∈ [0, 1] there is a finite semi-split decomposition series for ker π_{t} of length four whose succesive quotients are either semi-contractible or crossed-contractible.

As a final step we can also remove the stabilizations by introducing a little longer decompostion series and slightly more general succesive quotients. When μ : A → B is a ∗-homomorphism between separable C^{∗}-algebras which is an isomorphism in KK, the mapping cone {(a, f) ∈ A ⊕ cone(B) : f(1) = μ(a)} is KK-contractible. For our purposes here we need only consider the very special case where B = A⊗K and the ∗-homomorphism is the canonical stabilizing ∗-homomorphism s : A → A ⊗ K. Since s is injective we can consider A as a C^{∗}-subalgebra of A ⊗ K and the mapping cone can be described as

C_{A} = {f ∈ cone(A ⊗ K) : f(1) ∈ A} .

Set A = \{f ∈ C([0, 1], A ⊗ K) : f(t) ∈ A, t ≥ \frac{1}{2}\} and let π_{t} be evaluation at t ∈ [0, 1]. Then (A, [0, 1], π) is a semi-split and piecewise trivial continuous bundle of C^{∗}-algebras connecting A ⊗ K to A. For each t ∈ [0, 1], ker π_{t} is a semi-split extension, either of a contractible C^{∗}-algebra by a contractible C^{∗}-algebra, or of C_{A} by a contractible C^{∗}-algebra.

Theorem 7.8. Let A and B be separable C^{∗}-algebras. Then A and B are KKequivalent if and only if there is a separable semi-split and piecewise trivial continuous bundle of C^{∗}-algebras, (A, [0, 1], π), with no more than six points of nontriviality, such that π_{0}(A) ≃ A, π_{1}(A) ≃ B, and for each t ∈ [0, 1] there is a finite semi-split decomposition series for ker π_{t} of length eight whose succesive quotients are either semi-contractible, crossed-contractible or isomorphic to C_{A} or C_{B}.

Proof. Concatenate the bundles just described, one for A and one for B, with the bundle from Theorem 7.7. Apply Lemma 7.2 and Lemma 7.3 to prove the ‘only if’ part. For the ‘if’ part observe that since the bundle is semi-split the extensions (8) are semi-split. Furthermore, ker π_{t} is contractible in KK-theory since its decomposition series is semi-split and the succesive quotients are all KKcontractible; the semi-contractible quotients by Theorem 3.2. Since KK-theory is half-exact with respect to semi-split extensions by [K], [CS], it follows that π_{t} is a KK-equivalence and that A_{t} is KK-equivalent to A for all t. □

Corollary 7.9. Let A and B be separable C^{∗}-algebras. Then A and B are KKequivalent if and only if there is a separable C^{∗}-algebra D and surjective semi-split ∗-homomorphisms φ : D → A and ψ : D → B such that ker φ and ker ψ are KK-contractible.

References

[A] W. Arveson, Notes on extensions of C^{∗}-algebras, Duke Math. J. 44 (1977), 329–355, MR 55 #11056, Zbl 368.46052.

[BK] B. Blackadar, E. Kirchberg, Generalized inductive limits of finite-dimensional C^{∗}-algebras, Math. Ann. 307 (1997), 343–380, MR 98c:46112, Zbl 874.46036.

[CE] M.-D. Choi, E. Effros, The completely positive lifting problem for C^{∗}-algebras, Ann. of Math. 104 (1976), 585–609, MR 54 #5843, Zbl 361.46067.

[CH] A.Connes, N. Higson, Déformations, morphismes asymptotiques et K-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 101–106, MR 91m:46114, Zbl 717.46062.

[CS] J. Cuntz, G. Skandalis, Mapping cones and exact sequences in KK-theory, J. Operator Theory 15 (1986), 163–180, MR 88b:46099, Zbl 603.46063.

[D] M. Dadarlat, Shape theory and asymptotic morphisms for C^{∗}-algebras, Duke Math. J. 73 (1997), 687–711, MR 95c:46117, Zbl 847.46028.

306 Klaus Thomsen

[E] G.A. Elliott, On the classification of C^{∗}-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179–219, MR 94i:46074, Zbl 809.46067.

[ENN] G.A. Elliott, T. Natsume, R. Nest, The Heisenberg group and K-theory, K-theory 7 (1993), 409–428, MR 94m:46109, Zbl 803.46076.

[FS] T. Fack, G. Skandalis, Connes’ analogue of the Thom isomorphism for the Kasparov groups, Inventiones Math. 64 (1981), 7–14, MR 82g:46113, Zbl 482.46043.

[K] G.G. Kasparov, The operator K-functor and extensions of C^{∗}-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 571–636, MR 81m:58075, Zbl 448.46051.

[KW] E. Kirchberg, S. Wassermann, Operations on continuous bundles of C^{∗}-algebras, Math. Ann. 303 (1995), 677–697, MR 96j:46057, Zbl 835.46057.

[L] T. Loring, A test for injectivity for asymptotic morphisms, in Algebraic Methods in Operator Theory (R. Curto and P. Jorgensen, eds.) Birkhäuser, Boston, 1991, pp. 272–275, MR 95h:46110, Zbl 814.46046.

[MT] V.M. Manuilov, K. Thomsen, Asymptotically split extensions and E-theory, Algebra i Analiz (St. Petersburg Math. J.), to appear.

[R] M. Rieffel, Continuous fields of C^{∗}-algebras coming from group cycles and actions, Math. Ann. 283 (1989), 631–643, MR 90b:46120, Zbl 646.46063.

[RS] J. Rosenberg, C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (1987), 431–474, MR 88i:46091, Zbl 644.46051.

[S] G. Skandalis, Le bifoncteur de Kasparov n’est pas exact , C. R. Acad. Sci. Paris, 313 (1991), 939–941, MR 93b:46136, Zbl 744.46066.

[Th1] K. Thomsen, Discrete asymptotic homomorphisms in E-theory and KK-theory, Preprint, 1998.

[Th2] K. Thomsen , Homotopy invariance for bifunctors defined from asymptotic homomorphisms, Preprint, 1999.

Institut for matematiske fag, Ny Munkegade, 8000 Aarhus C, Denmark matkt@imf.au.dk

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