The dynamical Kirchberg-Phillips theorem

TL;DR

This paper develops a classification theory for amenable group actions on Kirchberg algebras, extending to all locally compact groups and solving longstanding conjectures in the field.

Contribution

It introduces a new classification framework for amenable G-actions on Kirchberg algebras using equivariant Kasparov theory, applicable to arbitrary locally compact groups.

Findings

Classifies G-C*-dynamical systems via equivariant Kasparov theory.

Solves Izumi's conjecture for discrete amenable torsion-free groups.

Recovers main results for poly-Z group actions.

Abstract

Let $G$ be a second-countable, locally compact group. In this article we study amenable $G$-actions on Kirchberg algebras that admit an approximately central embedding of a canonical quasi-free action on the Cuntz algebra $\mathcal{O}_\infty$. If $G$ is discrete, this coincides with the class of amenable and outer $G$-actions on Kirchberg algebras. We show that the resulting $G$-C*-dynamical systems are classified by equivariant Kasparov theory up to cocycle conjugacy. This is the first classification theory of its kind applicable to actions of arbitrary locally compact groups. Among various applications, our main result solves a conjecture of Izumi for actions of discrete amenable torsion-free groups, and recovers the main results of recent work by Izumi-Matui for actions of poly-$\mathbb{Z}$ groups.