On the KK-theory of strongly self-absorbing C^*-algebras - Dadarlat, Marius; Winter, Wilhelm (MATHEMATICA SCANDINAVICA, 2009)
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 K_1-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, AD) in terms of ^*-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group KK(D, AD) is isomorphic to K_0(AD).