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

Abstract. Let D and A be unital and separable C∗-algebras; let D be strongly self-absorbing. 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 en-domorphism 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 re-spective statement holds for the space of unital endomorphisms of D. As an application, we give a description of the Kasparov groupKK(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). A unital and separable C∗-algebra D 6 = C is strongly self-absorbing if there is an isomorphism D ∼ → D ⊗ D which is approximately unitarily equivalent to the inclusion map D → D ⊗ D, d 7 → d ⊗ 1D ([14]). Strongly self-absorbing C∗-algebras are known t