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

Abstract

Let $\Dh$ and $A$ be unital and separable $C^{\ast }$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital ${}^{\ast }$-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 ${}^{\ast }$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.

12 pages

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