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$^*$-algebras and there is a unital embedding $A \rightarrow B$ which is invertible on $KK$-theory and traces, then $A \cong B$. In particular, two unital separable simple nuclear $\mathcal Z$-stable C$^*$-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$^*$-algebras are isomorphic if and only if they are $KK$-equivalent in a unit-preserving way.