KK-rigidity of simple nuclear C*-algebras

Abstract

It is shown that if $A$ and $B$ are unital separable simple nuclear $\mathcal{Z}$-stable C${}^{\ast }$-algebras and there is a unital embedding $A\to B$ which is invertible on $KK$-theory and traces, then $A\cong B$. In particular, two unital separable simple nuclear $\mathcal{Z}$-stable C${}^{\ast }$-algebras which either have real rank zero or unique trace are isomorphic if and only if they are homotopy equivalent. It is further shown that two finite strongly self-absorbing C${}^{\ast }$-algebras are isomorphic if and only if they are $KK$-equivalent in a unit-preserving way.

The previous version had a gap in the proof of Theorem A coming from a mistake in a previous paper. Relatively minor modifications have been made to account for this. The main results have not changed. 16 pages