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

Let $\text{Dh}$ and $A$ be unital and separable $C^{\ast }$-algebras; let $\text{Dh}$ be strongly self-absorbing. It is known that any two unital ${}^{\ast }$-homomorphisms from $\text{Dh}$ to $A\otimes \text{Dh}$ are approximately unitarily equivalent. We show that, if $\text{Dh}$ is also $K_1$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\text{Dh}$ is asymptotically inner. Moreover, the space of automorphisms of $\text{Dh}$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\text{Aut}(\text{Dh})]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\text{Dh}$. As an application, we give a description of the Kasparov group $KK(\text{Dh},A\text{ot}\text{Dh})$ in terms of ${}^{\ast }$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\text{Dh},A\text{ot}\text{Dh})$ is isomorphic to $K_0(A\text{ot}\text{Dh})$.