KK-rigidity of simple nuclear C*-algebras

TL;DR

This paper proves that under certain conditions, unital separable simple nuclear Z-stable C*-algebras are isomorphic if they are KK-theoretically and trace-wise invertible, establishing a rigidity result in classification theory.

Contribution

It establishes KK-rigidity for simple nuclear Z-stable C*-algebras and characterizes isomorphism via KK-equivalence for strongly self-absorbing C*-algebras.

Findings

Isomorphism of certain C*-algebras characterized by KK-theory and traces.

Unital separable simple nuclear Z-stable C*-algebras with real rank zero or unique trace are classified by homotopy.

Strongly self-absorbing C*-algebras are isomorphic iff they are KK-equivalent in a unital way.

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.