Equivariant Kirchberg-Phillips-type absorption for amenable group actions

TL;DR

This paper proves an equivariant absorption theorem for outer actions of amenable groups on Kirchberg algebras, extending known results and establishing a new classification result for such actions.

Contribution

It generalizes Kirchberg-Phillips-type absorption results to all amenable groups and introduces a homotopy rigidity classification for outer actions.

Findings

Established an equivariant absorption theorem for amenable group actions.

Proved that model actions are classified by natural properties up to strong cocycle conjugacy.

Demonstrated that homotopy equivalent outer actions are conjugate, enabling classification.

Abstract

We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C*-dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.