On the classification of group actions on \(C^*\)-algebras up to equivariant \(KK\)-equivalence (Q1982604)

The paper deals with the classification of group actions on \(C^*\)-algebras up to equivariant \(KK\)-equivalence. Let \(G\) be a second countable, locally compact group. It is shown that any group action is equivariantly \(KK\)-equivalent to an action on a simple, purely infinite \(C^*\)-algebra. If \(A\) is a Kirchberg algebra, it is shown that the \(KK^G\)-equivalence classes of actions of a torsion-free amenable group \(G\) on \(A\) are in one-to-one correspondence with the isomorphism classes of principal \(\operatorname{Aut}(A)\)-bundles over the classifying space \(BG\). For a cyclic group of prime order, its actions (not necessarily satisfying the Rokhlin property) are classified up to equivariant \(KK\)-equivalence using the Köhler invariant, in particular, the classification of \(\mathbb Z/p\) actions on stabilised Cuntz algebras in the equivariant bootstrap class is described.