On the Kk-theory of Strongly Self-absorbing C∗-algebras

Let D and A be unital and separable C∗-algebras; let D be strongly selfabsorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are approximately unitarily equivalent. We show that, if D is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of D is asymptotically inner. Moreover, the space of automorphisms of D 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 (D)] reduces to a point. The respective statement holds for the space of unital endomorphisms of D. As an application, we give a description of the Kasparov group KK(D, A⊗D) in terms of ∗-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group KK(D, A⊗D) is isomorphic to K0(A⊗D).