Equivariant property Gamma and the tracial local-to-global principle for C∗-dynamics

Szabó, Gábor; Wouters, Lise

doi:10.2140/apde.2025.18.1385

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Equivariant property Gamma and the tracial local-to-global principle for $\mathrm{C}^*$-dynamics

Gábor Szabó and Lise Wouters

Vol. 18 (2025), No. 6, 1385–1432

DOI: 10.2140/apde.2025.18.1385

Abstract

We consider the notion of equivariant uniform property Gamma for actions of countable discrete groups on $C^{*}$ -algebras that admit traces. In case the group is amenable and the $C^{*}$ -algebra has a compact tracial state space, we prove that this property implies a kind of tracial local-to-global principle for the $C^{*}$ -dynamical system, generalizing a recent such principle for $C^{*}$ -algebras exhibited in work of Castillejos et al. For actions on simple nuclear $𝒵$ -stable $C^{*}$ -algebras, we use this to prove that equivariant uniform property Gamma is equivalent to equivariant $𝒵$ -stability, generalizing a result of Gardella, Hirshberg, and Vaccaro.

Keywords

$\mathrm{C}^*$-dynamics, amenability, Jiang–Su algebra

Mathematical Subject Classification

Primary: 46L35, 46L55

Milestones

Received: 28 February 2023

Revised: 22 April 2024

Accepted: 26 June 2024

Published: 29 May 2025

Authors

Gábor Szabó
Department of Mathematics KU Leuven Leuven Belgium

Lise Wouters
Leuven Belgium

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Vol. 18, No. 6, 2025