Classification of equivariantly $\mathcal{O}_2$-stable amenable actions on nuclear $\mathrm{C}^\ast$-algebras

TL;DR

This paper classifies certain amenable group actions on nuclear C*-algebras using the induced action on the primitive ideal space, providing a comprehensive classification framework for various group types.

Contribution

It introduces a classification of equivariantly $\\mathcal{O}_2$-stable amenable actions on nuclear C*-algebras based on the induced primitive ideal space action, extending known results.

Findings

Actions are classified by the induced $G$-action on the primitive ideal space.

Unital classification theorem established for exact groups.

Classification up to conjugacy obtained for compact groups.

Abstract

Given a second-countable, locally compact group $G$, we consider amenable $G$-actions on separable, stable, nuclear $\mathrm{C}^\ast$-algebras that are isometrically shift-absorbing and tensorially absorb the trivial action on the Cuntz algebra $\mathcal{O}_2$. We show that such actions are classified up to cocycle conjugacy by the induced $G$-action on the primitive ideal space. In the special case when $G$ is exact, we prove a unital version of our classification theorem. For compact groups, we obtain a classification up to conjugacy.