The dynamical Kirchberg–Phillips theorem

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 $G-C^\ast$-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.

2010 Mathematics Subject Classification

19K35, 46L35, 46L55

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In memory of Eberhard Kirchberg

Received 23 May 2022

Accepted 8 June 2023

Published 10 May 2024