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

## TL;DR

This paper investigates the classification of group actions on C*-algebras using equivariant KK-theory, establishing key equivalences and classifications for actions of various groups, including cyclic groups of prime order.

## Contribution

It demonstrates that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra and relates a conjecture of Izumi to cocycle conjugacy and KK-equivalence.

## Key findings

- Any group action is KK-equivalent to an action on a simple, purely infinite C*-algebra.
- Classifies actions of cyclic groups of prime order on stabilized Cuntz algebras.
- Establishes the equivalence of Izumi's conjecture with cocycle conjugacy and KK-equivalence.

## Abstract

We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-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 K\"ohler. In particular, we classify actions of G on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.