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

Abstract

Let and A be unital and separable C* -algebras; let be strongly self-absorbing. It is known that any two unital -homomorphisms from to A ⊗ are approximately unitarily equivalent. We show that, if is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of is asymptotically inner. Moreover, the space of automorphisms of 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( )] reduces to a point. The respective statement holds for the space of unital endomorphisms of . As an application, we give a description of the Kasparov group KK(, A ⊗ ) in terms of *-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group KK(, A ⊗ ) is isomorphic to K 0(A ⊗ ).