On the classification of group actions on C*-algebras up to equivariant KK-equivalence

Abstract

We study the classification of group actions on ${\text{C}}^{\ast }$-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite ${\text{C}}^{\ast }$-algebra. We show that a conjecture of Izumi is equivalent to an equivalence between cocycle conjugacy and equivariant KK-equivalence for actions of torsion-free amenable groups on Kirchberg algebras. Let $G$ be a cyclic group of prime order. We describe its actions up to equivariant KK-equivalence, based on previous work by Manuel Köhler. In particular, we classify actions of $G$ on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.