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

TL;DR

This paper introduces an equivariant version of property Gamma for group actions on C*-algebras with traces, establishing its equivalence to Z-stability under certain conditions and extending local-to-global principles in C*-dynamics.

Contribution

It defines equivariant uniform property Gamma for C*-dynamical systems and proves its equivalence to Z-stability for actions on simple nuclear Z-stable C*-algebras.

Findings

Equivariant property Gamma implies a tracial local-to-global principle.

Equivariant property Gamma is equivalent to Z-stability for certain C*-dynamics.

Generalizes recent results on local-to-global principles and stability in C*-algebras.

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 $\mathcal{Z}$-stable C*-algebras, we use this to prove that equivariant uniform property Gamma is equivalent to equivariant $\mathcal{Z}$-stability, generalizing a result of Gardella-Hirshberg-Vaccaro.