On the KK-theory of strongly self-absorbing C*-algebras

TL;DR

This paper explores the KK-theory of strongly self-absorbing C*-algebras, establishing asymptotic unitary equivalence of certain homomorphisms, describing automorphism spaces, and relating KK-groups to K-theory.

Contribution

It proves that unital endomorphisms of strongly self-absorbing C*-algebras are asymptotically inner and characterizes their automorphism spaces and KK-theory in new ways.

Findings

Unital endomorphisms are asymptotically inner.

Automorphism spaces are compactly-contractible.

KK(, A) is isomorphic to K_0(A).

Abstract

Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\Aut(\Dh)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\Dh$. As an application, we give a description of the Kasparov group $KK(\Dh, A\ot \Dh)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.

Equations

$$\Vert u_n\sigma (a)u_n^{\ast }-\gamma (a)\Vert \stackrel{n\to \infty }{⟶}0$$

$$\Vert u_t\sigma (a)u_t^{\ast }-\gamma (a)\Vert \stackrel{t\to \infty }{⟶}0$$

$$\Vert [w,\sigma (d)]\Vert <\delta$$

$$\Vert [w_t,\sigma (d)]\Vert <\epsilon$$

$$\Vert u(d\otimes 1_{\mathcal{D}})u^{\ast }-1_{\mathcal{D}}\otimes d\Vert <\frac{\epsilon }{20}$$

$$\Vert u-\sum _{i=1}^k{s}_i\otimes t_i\Vert <\frac{\epsilon }{20}.$$

$$\delta :=\frac{\epsilon }{k\cdot 10}$$

$$\mathcal{G}:=\{s_1,\dots ,s_k\}\subset \mathcal{D}.$$

$$\Vert [w,\sigma (s_i)]\Vert <\delta$$

$$\phi :A\otimes \mathcal{D}\otimes \mathcal{D}\to A\otimes \mathcal{D}$$

$$\Vert \phi (a\otimes 1_{\mathcal{D}})-a\Vert <\frac{\epsilon }{20}$$

$${\bar{w}}_{\frac{1}{2}}=w\text{mbox}and{\bar{w}}_1=1_{A\otimes \mathcal{D}}.$$

$$w_t:=\phi ((\sigma \otimes {\mathrm{id}}_{\mathcal{D}})(u{)}^{\ast }({\bar{w}}_t\otimes 1_{\mathcal{D}})(\sigma \otimes {\mathrm{id}}_{\mathcal{D}})(u))\in \mathcal{U}(A\otimes \mathcal{D});$$

$$\Vert [w_t,\sigma (d)]\Vert$$

$$\Vert w_{\frac{1}{2}}-w\Vert$$

$$w_t:=(\exp (2\pi ita))w\in \mathcal{U}(A\otimes \mathcal{D}).$$

$$\Vert w_t-w\Vert \le \Vert w_{\frac{1}{2}}-w\Vert <\frac{\epsilon }{3}$$

$$\Vert [w_t,\sigma (d)]\Vert <\Vert [w_{\frac{1}{2}},\sigma (d)]\Vert +\frac{2}{3}\epsilon \stackrel{\text{eqref}9}{<}\epsilon$$

$${\mathcal{F}}_0\subset {\mathcal{F}}_1\subset \dots$$

$${\mathcal{F}}_n\subset {\mathcal{G}}_n\subset {\mathcal{G}}_{n+1}\text{mbox}andthat{\delta }_{n+1}<{\delta }_n<{\epsilon }_n$$

$$\Vert u_n\sigma (d)u_n^{\ast }-\gamma (d)\Vert <\frac{{\delta }_n}{2}$$

$$w_n:=u_{n+1}^{\ast }{u}_n,n\in \mathbb{N}.$$

$$\Vert [w_n,\sigma (d)]\Vert$$

$$\Vert [w_{n,t},\sigma (d)]\Vert <{\epsilon }_n$$

$${\bar{u}}_t:=u_{n+1}{w}_{n,t-n}\text{mbox}ift\in [n,n+1).$$

$${\bar{u}}_n=u_{n+1}{w}_n=u_n$$

$${\bar{u}}_t\to u_{n+1}$$

$$\Vert {\bar{u}}_t\sigma (d){\bar{u}}_t^{\ast }-\gamma (d)\Vert$$

$$\Vert {\bar{u}}_t\sigma (d){\bar{u}}_t^{\ast }-\gamma (d)\Vert \stackrel{t\to \infty }{⟶}0$$

$$K{K}_{i+1}(C_{\nu }\mathcal{D},A\otimes \mathcal{D})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

Supported by: The first named author was partially supported by NSF grant #DMS-0500693.

The second named author was supported by the DFG (SFB 478).

0. Introduction

A unital and separable $C^{*}$-algebra ${\mathcal{D}}\neq{\mathbb{C}}$ is strongly self-absorbing if there is an isomorphism ${\mathcal{D}}\stackrel{{\scriptstyle\sim}}{{\to}}{\mathcal{D}}\otimes{\mathcal{D}}$ which is approximately unitarily equivalent to the inclusion map ${\mathcal{D}}\to{\mathcal{D}}\otimes{\mathcal{D}}$, $d\mapsto d\otimes\mathbf{1}_{{\mathcal{D}}}$ ([14]). Strongly self-absorbing $C^{*}$-algebras are known to be simple and nuclear; moreover, they are either purely infinite or stably finite. The only known examples of strongly self-absorbing $C^{*}$-algebras are the UHF algebras of infinite type (i.e., every prime number that occurs in the respective supernatural number occurs with infinite multiplicity), the Cuntz algebras ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{\infty}$, the Jiang–Su algebra ${\mathcal{Z}}$ and tensor products of ${\mathcal{O}}_{\infty}$ with UHF algebras of infinite type, see [14]. All these examples are $K_{1}$-injective, i.e., the canonical map ${\mathcal{U}}({\mathcal{D}})/{\mathcal{U}}_{0}({\mathcal{D}})\to K_{1}({\mathcal{D}})$ is injective.

It was observed in [14] that any two unital ∗-homomorphisms $\sigma,\gamma:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ are approximately unitarily equivalent, were $A$ is another unital and separable $C^{*}$-algebra. If ${\mathcal{D}}$ is $K_{1}$-injective, the unitaries implementing the equivalence may even be chosen to be homotopic to the unit. When ${\mathcal{D}}$ is ${\mathcal{O}}_{2}$, ${\mathcal{O}}_{\infty}$, it was known that $\sigma$ and $\gamma$ are even asymptotically unitarily equivalent – i.e., they can be intertwined by a continuous path of unitaries, parametrized by a half-open interval. Up to this point, it was not clear whether the respective statement holds for the Jiang–Su algebra ${\mathcal{Z}}$. Theorem 2 below provides an affirmative answer to this problem. Even more, we show that the path intertwining $\sigma$ and $\gamma$ may be chosen in the component of the unit.

We believe this result, albeit technical, is interesting in its own right, and that it will be a useful ingredient for the systematic further use of strongly self-absorbing $C^{*}$-algebras in Elliott’s program to classify nuclear $C^{*}$-algebras by $K$-theory data. In fact, this point of view is our main motivation for the study of strongly self-absorbing $C^{*}$-algebras; see [8], [10], [16], [17], [18] and [15] for already existing results in this direction.

For the time being, we use Theorem 2 to derive some consequences for the Kasparov groups of the form $KK({\mathcal{D}},A\otimes{\mathcal{D}})$. More precisely, we show that all the elements of the Kasparov group $KK({\mathcal{D}},A\otimes{\mathcal{D}})$ are of the form $[\varphi]-n[\iota]$ where $\varphi:{\mathcal{D}}\to\mathcal{K}\otimes A\otimes{\mathcal{D}}$ is a ∗-homomorphism and $\iota:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ is the inclusion $\iota(d)=\mathbf{1}_{A}\otimes d$ and $n\in\mathbb{N}$. Moreover, two non-zero ∗-homomorphisms $\varphi,\psi:{\mathcal{D}}\to\mathcal{K}\otimes A\otimes{\mathcal{D}}$ with $\varphi(\mathbf{1}_{\mathcal{D}})=\psi(\mathbf{1}_{\mathcal{D}})=e$ have the same KK-theory class if and only if there is a unitary-valued continuous map $u:[0,1)\to e(\mathcal{K}\otimes A\otimes{\mathcal{D}})e$, $t\mapsto u_{t}$ such that $u_{0}=e$ and $\lim_{t\to 1}\|u_{t}\,\varphi(d)\,u_{t}^{*}-\psi(d)\|=0$ for all $d\in{\mathcal{D}}$. In addition, we show that $KK_{i}({\mathcal{D}},{\mathcal{D}}\otimes A)\cong K_{i}({\mathcal{D}}\otimes A)$, $i=0,1$.

One may note the similarity to the descriptions of $KK(\mathcal{O}_{\infty},\mathcal{O}_{\infty}\otimes A)$ ([8],[10]) and $KK(\mathbb{C},\mathbb{C}\otimes A)$. However, we do not require that ${\mathcal{D}}$ satisfies the universal coefficient theorem (UCT) in KK-theory. In the same spirit, we characterize ${\mathcal{O}}_{2}$ and the universal UHF algebra ${\mathcal{Q}}$ using $K$-theoretic conditions, but without involving the UCT.

As another application of Theorem 2 (and the results of [7]), we prove in [4] an automatic trivialization result for continuous fields with strongly self-absorbing fibres over finite dimensional spaces.

The second named author would like to thank Eberhard Kirchberg for an inspiring conversation on the problem of proving Theorem 2.

1. Strongly self-absorbing $C^{*}$-algebras

In this section we recall the notion of strongly self-absorbing $C^{*}$-algebras and some facts from [14].

1.1 Definition: * Let $A$, $B$ be $C^{*}$-algebras and $\sigma,\gamma:A\to B$ be ∗-homomorphisms. Suppose that $B$ is unital.*

1. (i)

We say that $\sigma$ and $\gamma$ are approximately unitarily equivalent, $\sigma\approx_{\mathrm{u}}\gamma$, if there is a sequence $(u_{n})_{n\in{\mathbb{N}}}$ of unitaries in $B$ such that

[TABLE]

for every $a\in A$. If all $u_{n}$ can be chosen to be in ${\mathcal{U}}_{0}(B)$, the connected component of $\mathbf{1}_{B}$ of the unitary group ${\mathcal{U}}(B)$, then we say that $\sigma$ and $\gamma$ are strongly approximately unitarily equivalent, written $\sigma\approx_{\mathrm{su}}\gamma$. 2. (ii)

We say that $\sigma$ and $\gamma$ are asymptotically unitarily equivalent, $\sigma\approx_{\mathrm{uh}}\gamma$, if there is a norm-continuous path $(u_{t})_{t\in[0,\infty)}$ of unitaries in $B$ such that

[TABLE]

for every $a\in A$. If one can arrange that $u_{0}=\mathbf{1}_{B}$ and hence ($u_{t}\in{\mathcal{U}}_{0}(B)$* for all $t$), then we say that $\sigma$ and $\gamma$ are strongly asymptotically unitarily equivalent, written $\sigma\approx_{\mathrm{suh}}\gamma$.*

1.2 The concept of strongly self-absorbing $C^{*}$-algebras was formally introduced in [14, Definition 1.3]:

Definition: *

A separable unital $C^{*}$-algebra ${\mathcal{D}}$ is strongly self-absorbing, if ${\mathcal{D}}\neq{\mathbb{C}}$ and there is an isomorphism $\varphi:{\mathcal{D}}\to{\mathcal{D}}\otimes{\mathcal{D}}$ such that $\varphi\approx_{\mathrm{u}}\mathrm{id}_{{\mathcal{D}}}\otimes\mathbf{1}_{{\mathcal{D}}}$. *

1.3 Recall [14, Corollary 1.12]:

Proposition: * Let $A$ and ${\mathcal{D}}$ be unital $C^{*}$-algebras, with ${\mathcal{D}}$ strongly self-absorbing. Then, any two unital ∗-homomorphisms $\sigma,\gamma:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ are approximately unitarily equivalent. In particular, any two unital endomorphisms of ${\mathcal{D}}$ are approximately unitarily equivalent. *

We note that the assumption that $A$ is separable which appears in the original statement of [14, Corollary 1.12] is not necessary and was not used in the proof.

1.4 Lemma: * Let ${\mathcal{D}}$ be a strongly self-absorbing $C^{*}$-algebra. Then there is a sequence of unitaries $(w_{n})_{n\in{\mathbb{N}}}$ in the commutator subgroup of ${\mathcal{U}}({\mathcal{D}}\otimes{\mathcal{D}})$ such that for all $d\in{\mathcal{D}}$ $\|w_{n}(d\otimes\mathbf{1}_{\mathcal{D}})w_{n}^{*}-\mathbf{1}_{\mathcal{D}}\otimes d\|\to 0$ as $n\to\infty$. *

Proof: Let ${\mathcal{F}}\subset{\mathcal{D}}$ be a finite normalized set and let $\varepsilon>0$. By [14, Prop. 1.5] there is a unitary $u\in{\mathcal{U}}({\mathcal{D}}\otimes{\mathcal{D}})$ such that $\|u(d\otimes\mathbf{1}_{{\mathcal{D}}})u^{*}-\mathbf{1}_{{\mathcal{D}}}\otimes d\|<\varepsilon$ for all $d\in{\mathcal{F}}$. Let $\theta:{\mathcal{D}}\otimes{\mathcal{D}}\to{\mathcal{D}}$ be a ∗-isomorphism. Then $\|(\theta(u^{*})\otimes\mathbf{1}_{{\mathcal{D}}})u(d\otimes\mathbf{1}_{{\mathcal{D}}})u^{*}(\theta(u)\otimes\mathbf{1}_{{\mathcal{D}}})-\mathbf{1}_{{\mathcal{D}}}\otimes d\|<\varepsilon$ for all $d\in{\mathcal{F}}$. By Proposition 1 $\theta\otimes\mathbf{1}_{\mathcal{D}}\approx_{\mathrm{u}}\mathrm{id}_{{\mathcal{D}}\otimes{\mathcal{D}}}$ and so there is a unitary $v\in{\mathcal{U}}({\mathcal{D}}\otimes{\mathcal{D}})$ such that $\|\theta(u^{*})\otimes\mathbf{1}_{\mathcal{D}}-vu^{*}v^{*}\|<\varepsilon$ and hence $\|(\theta(u^{*})\otimes\mathbf{1}_{\mathcal{D}})u-vu^{*}v^{*}u\|<\varepsilon$. Setting $w=vu^{*}v^{*}u$ we deduce that $\|w(d\otimes\mathbf{1}_{{\mathcal{D}}})w^{*}-\mathbf{1}_{{\mathcal{D}}}\otimes d\|<3\varepsilon$ for all $d\in{\mathcal{F}}$.

1.5 Remark: In the situation of Proposition 1, suppose that the commutator subgroup of ${\mathcal{U}}({\mathcal{D}})$ is contained in ${\mathcal{U}}_{0}({\mathcal{D}})$. This will happen for instance if ${\mathcal{D}}$ is assumed to be $K_{1}$-injective. Then one may choose the unitaries $(u_{n})_{n\in{\mathbb{N}}}$ which implement the approximate unitary equivalence between $\sigma$ and $\gamma$ to lie in ${\mathcal{U}}_{0}(A\otimes{\mathcal{D}})$. This follows from [14, (the proof of) Corollary 1.12], since the unitaries $(u_{n})_{n\in{\mathbb{N}}}$ are essentially images of the unitaries $(w_{n})_{n\in{\mathbb{N}}}$ of Lemma 1 under suitable unital ∗-homomorphisms.

2. Asymptotic vs. approximate unitary equivalence

It is the aim of this section to establish a continuous version of Proposition 1.

2.1 Lemma: * Let ${\mathcal{D}}$ be separable unital strongly self-absorbing $C^{*}$-algebra. For any finite subset ${\mathcal{F}}\subset{\mathcal{D}}$ and $\varepsilon>0$, there are a finite subset ${\mathcal{G}}\subset{\mathcal{D}}$ and $\delta>0$ such that the following holds:*

If $A$ is another unital $C^{*}$-algebra and $\sigma:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ is a unital ∗-homomorphism, and if $w\in{\mathcal{U}}_{0}(A\otimes{\mathcal{D}})$ is a unitary satisfying

[TABLE]

for all $d\in{\mathcal{G}}$, then there is a continuous path $(w_{t})_{t\in[0,1]}$ of unitaries in ${\mathcal{U}}_{0}(A\otimes{\mathcal{D}})$ such that $w_{0}=w$, $w_{1}=\mathbf{1}_{A\otimes{\mathcal{D}}}$ and

[TABLE]

*for all $d\in{\mathcal{F}}$, $t\in[0,1]$. *

Proof: We may clearly assume that the elements of ${\mathcal{F}}$ are normalized and that $\varepsilon<1$. Let $u\in{\mathcal{D}}\otimes{\mathcal{D}}$ be a unitary satisfying

[TABLE]

for all $d\in{\mathcal{F}}$. There exist $k\in{\mathbb{N}}$ and elements $s_{1},\ldots,s_{k},t_{1},\ldots,t_{k}\in{\mathcal{D}}$ of norm at most one such that

[TABLE]

Set

[TABLE]

and

[TABLE]

Now let $w\in{\mathcal{U}}_{0}(A\otimes{\mathcal{D}})$ be a unitary as in the assertion of the lemma, i.e., $w$ satisfies

[TABLE]

for all $i=1,\dots,k$. We proceed to construct the path $(w_{t})_{t\in[0,1]}$.

By [14, Remark 2.7] there is a unital ∗-homomorphism

[TABLE]

such that

[TABLE]

for all $a\in\sigma({\mathcal{F}})\cup\{w\}$.

Since $w\in{\mathcal{U}}_{0}(A\otimes{\mathcal{D}})$, there is a path $(\bar{w}_{t})_{t\in[\frac{1}{2},1]}$ of unitaries in $A\otimes{\mathcal{D}}$ such that

[TABLE]

For $t\in[\frac{1}{2},1]$ define

[TABLE]

then $(w_{t})_{t\in[\frac{1}{2},1]}$ is a continuous path of unitaries in $A\otimes{\mathcal{D}}$. For $t\in[\frac{1}{2},1]$ and $d\in{\mathcal{F}}$ we have

[TABLE]

where for the last equality we have used that the $\bar{w}_{t}$ are unitaries and that $\sigma$ is a unital ∗-homomorphism. Furthermore, we have

[TABLE]

The above estimate allows us to extend the path $(w_{t})_{t\in[\frac{1}{2},1]}$ to the whole interval $[0,1]$ in the desired way: We have $\|w_{\frac{1}{2}}w^{*}-\mathbf{1}_{{\mathcal{D}}}\|<\frac{\varepsilon}{3}<2$, whence $-1$ is not in the spectrum of $w_{\frac{1}{2}}w^{*}$. By functional calculus, there is $a=a^{*}\in A\otimes{\mathcal{D}}$ with $\|a\|<1$ such that $w_{\frac{1}{2}}w^{*}=\exp(\pi ia)$. For $t\in[0,\frac{1}{2})$ we may therefore define a continuous path of unitaries

[TABLE]

It is clear that $w_{0}=w$ and $w_{t}\to w_{\frac{1}{2}}$ as $t\to(\frac{1}{2})_{-}$, whence $(w_{t})_{t\in[0,1]}$ is a continuous path of unitaries in $A$ satisfying $w_{0}=w$ and $w_{1}=\mathbf{1}_{A}\otimes{\mathcal{D}}$. Moreover, it is easy to see that

[TABLE]

for all $t\in[0,\frac{1}{2})$, whence

[TABLE]

for $t\in[0,\frac{1}{2})$, $d\in{\mathcal{F}}$.

We have now constructed a path $(w_{t})_{t\in[0,1]}\subset{\mathcal{U}}(A)$ with the desired properties.

2.2 Theorem: * Let $A$ and ${\mathcal{D}}$ be unital $C^{*}$-algebras, with ${\mathcal{D}}$ separable, strongly self-absorbing and $K_{1}$-injective. Then, any two unital ∗-homomorphisms $\sigma,\gamma:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ are strongly asymptotically unitarily equivalent. In particular, any two unital endomorphisms of ${\mathcal{D}}$ are strongly asymptotically unitarily equivalent. *

Proof: Note that the second statement follows from the first one with $A={\mathcal{D}}$, since ${\mathcal{D}}\cong{\mathcal{D}}\otimes{\mathcal{D}}$ by assumption.

Let $A$ be a unital $C^{*}$-algebra such that $A\cong A\otimes{\mathcal{D}}$ and let $\sigma,\gamma:{\mathcal{D}}\to A$ be unital ∗-homomorphisms. We shall prove that $\sigma$ and $\gamma$ are strongly asymptotically unitarily equivalent. Choose an increasing sequence

[TABLE]

of finite subsets of ${\mathcal{D}}$ such that $\bigcup{\mathcal{F}}_{n}$ is a dense subset of ${\mathcal{D}}$. Let $1>\varepsilon_{0}>\varepsilon_{1}>\ldots$ be a decreasing sequence of strictly positive numbers converging to [math].

For each $n\in{\mathbb{N}}$, employ Lemma 2 (with ${\mathcal{F}}_{n}$ and $\varepsilon_{n}$ in place of ${\mathcal{F}}$ and $\varepsilon$) to obtain a finite subset ${\mathcal{G}}_{n}\subset{\mathcal{D}}$ and $\delta_{n}>0$. We may clearly assume that

[TABLE]

for all $n\in{\mathbb{N}}$.

Since $\sigma$ and $\gamma$ are strongly approximately unitarily equivalent by Proposition 1 and Remark 1, there is a sequence of unitaries $(u_{n})_{n\in{\mathbb{N}}}\subset{\mathcal{U}}_{0}(A)$ such that

[TABLE]

for all $d\in{\mathcal{G}}_{n}$ and $n\in{\mathbb{N}}$. Let us set

[TABLE]

Then $w_{n}\in{\mathcal{U}}_{0}(A)$ and

[TABLE]

for $d\in{\mathcal{G}}_{n}$, $n\in{\mathbb{N}}$. Now by Lemma 2 (and the choice of the ${\mathcal{G}}_{n}$ and $\delta_{n}$), for each $n$ there is a continuous path $(w_{n,t})_{t\in[0,1]}$ of unitaries in ${\mathcal{U}}_{0}(A)$ such that $w_{n,0}=w_{n}$, $w_{n,1}=\mathbf{1}_{A}$ and

[TABLE]

for all $d\in{\mathcal{F}}_{n}$, $t\in[0,1]$.

Next, define a path $(\bar{u}_{t})_{t\in[0,\infty)}$ of unitaries in ${\mathcal{U}}_{0}(A)$ by

[TABLE]

We have that

[TABLE]

and that

[TABLE]

as $t\to n+1$ from below, which implies that the path $(\bar{u}_{t})_{t\in[0,\infty)}$ is continuous in ${\mathcal{U}}_{0}(A)$. Furthermore, for $t\in[n,n+1)$ and $d\in{\mathcal{F}}_{n}$ we obtain

[TABLE]

Since the ${\mathcal{F}}_{n}$ are nested and the $\varepsilon_{n}$ converge to [math], we have

[TABLE]

for all $d\in\bigcup_{n=0}^{\infty}{\mathcal{F}}_{n}$; by continuity and since $\bigcup_{n=0}^{\infty}{\mathcal{F}}_{n}$ is dense in ${\mathcal{D}}$, we have (14) for all $d\in{\mathcal{D}}$. Since $\bar{u}_{0}\in{\mathcal{U}}_{0}(A)$ we may arrange that $\bar{u}_{0}=\mathbf{1}_{A}$.

3. The group $KK({\mathcal{D}},A\otimes{\mathcal{D}})$ and some applications

3.1 For a separable $C^{*}$-algebra ${\mathcal{D}}$ we endow the group of automorphisms $\mathrm{Aut}\,({\mathcal{D}})$ with the point-norm topology.

Corollary: * Let ${\mathcal{D}}$ be a separable, unital, strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra. Then $[X,\mathrm{Aut}({\mathcal{D}})]$ reduces to a point for any compact Hausdorff space $X$. *

Proof: Let $\varphi,\psi:X\to\mathrm{Aut}\,({\mathcal{D}})$ be continuous maps. We identify $\varphi$ and $\psi$ with unital ∗-homomorphisms $\varphi,\psi:{\mathcal{D}}\to{\mathcal{C}}(X)\otimes{\mathcal{D}}$. By Theorem 2, $\varphi$ is strongly asymptotically unitarily equivalent to $\psi$. This gives a homotopy between the two maps $\varphi,\psi:X\to\mathrm{Aut}\,({\mathcal{D}})$.

3.2 Remark: The conclusion of Corollary 3 was known before for ${\mathcal{D}}$ a UHF algebra of infinite type and $X$ a CW complex by [13], for ${\mathcal{D}}=\mathcal{O}_{2}$ by [8] and [10], and for ${\mathcal{D}}=\mathcal{O}_{\infty}$ by [2]. It is new for the Jiang–Su algebra.

3.3 For unital $C^{*}$-algebras ${\mathcal{D}}$ and $B$ we denote by $[{\mathcal{D}},B]$ the set of homotopy classes of unital ∗-homomorphisms from ${\mathcal{D}}$ to $B$. By a similar argument as above we also have the following corollary.

Corollary: * Let ${\mathcal{D}}$ and $A$ be unital $C^{*}$-algebras. If ${\mathcal{D}}$ is separable, strongly self-absorbing and $K_{1}$-injective, then $[{\mathcal{D}},A\otimes{\mathcal{D}}]$ reduces to a singleton. *

3.4 For separable unital $C^{*}$-algebras ${\mathcal{D}}$ and $B$, let $\chi_{i}:KK_{i}({\mathcal{D}},B)\to KK_{i}(\mathbb{C},B)\cong K_{i}(B)$, $i=0,1$ be the morphism of groups induced by the unital inclusion $\nu:\mathbb{C}\to{\mathcal{D}}$.

Theorem: * Let ${\mathcal{D}}$ be a unital, separable and strongly self-absorbing $C^{*}$-algebra. Then for any separable $C^{*}$-algebra $A$, the map $\chi_{i}:KK_{i}({\mathcal{D}},A\otimes{\mathcal{D}})\to K_{i}(A\otimes{\mathcal{D}})$ is bijective, for $i=0,1$. In particular both groups $KK_{i}({\mathcal{D}},A\otimes D)$ are countable and discrete with respect to their natural topology. *

Proof: Since ${\mathcal{D}}$ is KK-equivalent to ${\mathcal{D}}\otimes\mathcal{O}_{\infty}$, we may assume that ${\mathcal{D}}$ is purely infinite and in particular $K_{1}$-injective by [11, Prop. 4.1.4]. Let $C_{\nu}{\mathcal{D}}$ denote the mapping cone $C^{*}$-algebra of $\nu$. By [3, Cor. 3.10], there is a bijection $[{\mathcal{D}},A\otimes{\mathcal{D}}]\to KK(C_{\nu}{\mathcal{D}},SA\otimes{\mathcal{D}})$ and hence $KK(C_{\nu}{\mathcal{D}},SA\otimes{\mathcal{D}})=0$ for all separable and unital $C^{*}$-algebras $A$ as a consequence of Corollary 3. Since $KK(C_{\nu}{\mathcal{D}},A\otimes{\mathcal{D}})$ is isomorphic to $KK(C_{\nu}{\mathcal{D}},S^{2}A\otimes{\mathcal{D}})$ by Bott periodicity and the latter group injects in $KK(C_{\nu}{\mathcal{D}},SC(\mathbb{T})\otimes A\otimes{\mathcal{D}})=0$, we have that $KK_{i}(C_{\nu}{\mathcal{D}},{\mathcal{D}}\otimes A)=0$ for all unital and separable $C^{*}$-algebras $A$ and $i=0,1$. Since $KK_{i}(C_{\nu}{\mathcal{D}},{\mathcal{D}}\otimes A)$ is a subgroup of $KK_{i}(C_{\nu}{\mathcal{D}},{\mathcal{D}}\otimes\widetilde{A})=0$ (where $\widetilde{A}$ is the unitization of $A$) we see that $KK_{i}(C_{\nu}{\mathcal{D}},{\mathcal{D}}\otimes A)=0$ for all separable $C^{*}$-algebras $A$. Using the Puppe exact sequence, where $\chi_{i}=\nu^{*}$,

[TABLE]

we conclude that $\chi_{i}$ is an isomorphism, $i=0,1$. The map $\chi_{i}=\nu^{*}$ is continuous since it is given by the Kasparov product with a fixed element (we refer the reader to [12], [9] or [1] for a background on the topology of the Kasparov groups). Since the topology of $K_{i}$ is discrete and $\chi_{i}$ is injective, it follows that the topology of $KK_{i}({\mathcal{D}},A\otimes D)$ is also discrete. The countability of $KK_{i}({\mathcal{D}},A\otimes D)$ follows from that of $K_{i}(A\otimes D)$, as $A\otimes{\mathcal{D}}$ is separable.

3.5 Remark: In contrast to Theorem 3, if ${\mathcal{D}}$ is the universal UHF algebra, then $KK({\mathcal{D}},\mathbb{C})\cong\mathrm{Ext}(\mathbb{Q},\mathbb{Z})\cong\mathbb{Q}^{\mathbb{N}}$ has the power of the continuum [6, p. 221].

3.6 Let ${\mathcal{D}}$ and $A$ be as in Theorem 3 and assume in addition that ${\mathcal{D}}$ is $K_{1}$-injective and $A$ is unital. Let $\iota:{\mathcal{D}}\to A\otimes{\mathcal{D}}$ be defined by $\iota(d)=\mathbf{1}_{A}\otimes d$.

Corollary: * If $e\in\mathcal{K}\otimes A\otimes{\mathcal{D}}$ is a projection, and $\varphi,\psi:{\mathcal{D}}\to e(\mathcal{K}\otimes A\otimes{\mathcal{D}})e$ are two unital ∗-homomorphisms, then $\varphi\approx_{\mathrm{suh}}\psi$ and hence $[\varphi]=[\psi]\in KK({\mathcal{D}},A\otimes{\mathcal{D}})$. Moreover:*

[TABLE]

Proof: Let $\varphi$, $\psi$ and $e$ be as in the first part of the statement. By [14, Cor. 3.1], the unital $C^{*}$-algebra $e(\mathcal{K}\otimes A\otimes{\mathcal{D}})e$ is ${\mathcal{D}}$-stable, being a hereditary subalgebra of a ${\mathcal{D}}$-stable $C^{*}$-algebra. Therefore $\varphi\approx_{\mathrm{suh}}\psi$ by Theorem 2.

Now for the second part of the statement, let $x\in KK({\mathcal{D}},A\otimes{\mathcal{D}})$ be an arbitrary element. Then $\chi_{0}(x)=[e]-n[\mathbf{1}_{A\otimes{\mathcal{D}}}]$ for some projection $e\in\mathcal{K}\otimes A\otimes{\mathcal{D}}$ and $n\in\mathbb{N}$. Since $e(\mathcal{K}\otimes A\otimes{\mathcal{D}})e$ is ${\mathcal{D}}$-stable, there is a unital ∗-homomorphism $\varphi:{\mathcal{D}}\to e(\mathcal{K}\otimes A\otimes{\mathcal{D}})e$. Then

[TABLE]

and hence $[\varphi]-n[\iota]=x$ since $\chi_{0}$ is injective by Theorem 3.

In the remainder of the paper we give characterizations for the Cuntz algebra $\mathcal{O}_{2}$ and for the universal UHF-algebra which do not require the UCT. The latter result is a variation of a theorem of Effros and Rosenberg [5].

3.7 Proposition: * Let ${\mathcal{D}}$ be a separable unital strongly self-absorbing $C^{*}$-algebra. If $[\mathbf{1}_{{\mathcal{D}}}]=0$ in $K_{0}({\mathcal{D}})$, then ${\mathcal{D}}\cong\mathcal{O}_{2}$. *

Proof: Since ${\mathcal{D}}$ must be nuclear (see [14]), ${\mathcal{D}}$ embeds unitally in $\mathcal{O}_{2}$ by Kirchberg’s theorem. ${\mathcal{D}}$ is not stably finite since $[\mathbf{1}_{{\mathcal{D}}}]=0$. By the dichotomy of [14, Thm. 1.7] ${\mathcal{D}}$ must be purely infinite. Since $[\mathbf{1}_{{\mathcal{D}}}]=0$ in $K_{0}({\mathcal{D}})$, there is a unital embedding $\mathcal{O}_{2}\to{\mathcal{D}}$, see [11, Prop. 4.2.3]. We conclude that ${\mathcal{D}}$ is isomorphic to $\mathcal{O}_{2}$ by [14, Prop. 5.12].

3.8 Proposition: * Let ${\mathcal{D}}$, $A$ be separable, unital, strongly self-absorbing $C^{*}$-algebras. Suppose that for any finite subset $\mathcal{F}$ of ${\mathcal{D}}$ and any $\varepsilon>0$ there is a u.c.p. map $\varphi:{\mathcal{D}}\to A$ such that $\|\varphi(cd)-\varphi(c)\varphi(d)\|<\varepsilon$ for all $c,d\in\mathcal{F}$. Then $A\cong A\otimes{\mathcal{D}}$. *

Proof: By [14, Thm. 2.2] it suffices to show that for any given finite subsets $\mathcal{F}$ of ${\mathcal{D}}$, $\mathcal{G}$ of $A$ and any $\varepsilon>0$ there is u.c.p. map $\Phi:{\mathcal{D}}\to A$ such that (i) $\|\Phi(cd)-\Phi(c)\Phi(d)\|<\varepsilon$ for all $c,d\in\mathcal{F}$ and (ii) $\|[\Phi(d),a]\|<\varepsilon$ for all $d\in\mathcal{F}$ and $a\in\mathcal{G}$. We may assume that $\|d\|\leq 1$ for all $d\in\mathcal{F}$. Since $A$ is strongly self-absorbing, by [14, Prop. 1.10] there is a unital ∗-homomorphism $\gamma:A\otimes A\to A$ such that $\|\gamma(a\otimes\mathbf{1}_{A})-a\|<\varepsilon/2$ for all $a\in\mathcal{G}$. On the other hand, by assumption there is a u.c.p. map $\varphi:{\mathcal{D}}\to A$ such that $\|\varphi(cd)-\varphi(c)\varphi(d)\|<\varepsilon$ for all $c,d\in\mathcal{F}$. Let us define a u.c.p. map $\Phi:{\mathcal{D}}\to A$ by $\Phi(d)=\gamma(\mathbf{1}_{A}\otimes\varphi(d))$. It is clear that $\Phi$ satisfies (i) since $\gamma$ is a ∗-homomorphism. To conclude the proof we check now that $\Phi$ also satisfies (ii). Let $d\in\mathcal{F}$ and $a\in\mathcal{G}$. Then

[TABLE]

3.9 Proposition: * Let ${\mathcal{D}}$ be a separable, unital, strongly self-absorbing $C^{*}$-algebra. Suppose that ${\mathcal{D}}$ is quasidiagonal, it has cancellation of projections and that $[\mathbf{1}_{{\mathcal{D}}}]\in nK_{0}({\mathcal{D}})^{+}$ for all $n\geq 1$. Then ${\mathcal{D}}$ is isomorphic to the universal UHF algebra ${\mathcal{Q}}$ with $K_{0}({\mathcal{Q}})\cong\mathbb{Q}$. *

Proof: Since ${\mathcal{D}}$ is separable unital and quasidiagonal, there is a unital ∗-representation $\pi:{\mathcal{D}}\to B(H)$ on a separable Hilbert space $H$ and a sequence of nonzero projections $p_{n}\in B(H)$ of finite rank $k(n)$ such that $\lim_{n\to\infty}\|[p_{n},\pi(d)]\|=0$ for all $d\in{\mathcal{D}}$. Then the sequence of u.c.p. maps $\varphi_{n}:{\mathcal{D}}\to p_{n}B(H)p_{n}\cong M_{k(n)}(\mathbb{C})\subset{\mathcal{Q}}$ is asymptotically multiplicative, i.e $\lim_{n\to\infty}\|\varphi_{n}(cd)-\varphi_{n}(c)\varphi_{n}(d))\|=0$ for all $c,d\in{\mathcal{D}}$. Therefore ${\mathcal{Q}}\cong{\mathcal{Q}}\otimes{\mathcal{D}}$ by Proposition 3.

In the second part of the proof we show that ${\mathcal{D}}\cong{\mathcal{D}}\otimes{\mathcal{Q}}$. Let $E_{n}:{\mathcal{Q}}\to M_{n!}(\mathbb{C})\subset{\mathcal{Q}}$ be a conditional expectation onto $M_{n!}(\mathbb{C})$. Then $\lim_{n\to\infty}\|E_{n}(a)-a\|=0$ for all $a\in{\mathcal{Q}}$.

By assumption, for each $n$ there is a projection $e$ in ${\mathcal{D}}\otimes M_{m}(\mathbb{C})$ (for some $m$) such that $n![e]=[\mathbf{1}_{{\mathcal{D}}}]$ in $K_{0}({\mathcal{D}})$. Let $\varphi:M_{n!}(\mathbb{C})\to M_{n!}(\mathbb{C})\otimes e({\mathcal{D}}\otimes M_{m}(\mathbb{C}))e$ be defined by $\varphi(b)=b\otimes e$. Since ${\mathcal{D}}$ has cancellation of projections and since $n![e]=[\mathbf{1}_{{\mathcal{D}}}]$, there is a partial isometry $v\in M_{n!}(\mathbb{C})\otimes D\otimes M_{m}(\mathbb{C})$ such that $v^{*}v=\mathbf{1}_{M_{n!}(\mathbb{C})}\otimes e$ and $vv^{*}=e_{11}\otimes\mathbf{1}_{{\mathcal{D}}}\otimes e_{11}$. Therefore $b\mapsto v\,\varphi(b)\,v^{*}$ gives a unital embedding of $M_{n!}(\mathbb{C})$ into ${\mathcal{D}}$. Finally, $\psi_{n}(a)=v\,(\varphi\circ E_{n}(a))\,v^{*}$ defines a sequence of asymptotically multiplicative u.c.p. maps ${\mathcal{Q}}\to{\mathcal{D}}$. Therefore ${\mathcal{D}}\cong{\mathcal{D}}\otimes{\mathcal{Q}}$ by Proposition 3.

3.10 Remark: Let ${\mathcal{D}}$ be a separable, unital, strongly self-absorbing and quasidiagonal $C^{*}$-algebra. Then ${\mathcal{D}}\otimes{\mathcal{Q}}\cong{\mathcal{Q}}$ by the first part of the proof of Proposition 3. In particular $K_{1}({\mathcal{D}})\otimes\mathbb{Q}=0$ and $K_{0}({\mathcal{D}})\otimes\mathbb{Q}\cong\mathbb{Q}$ by the Künneth formula (or by writing ${\mathcal{Q}}$ as an inductive limit of matrices).