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

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.

47 pages, various small corrections and added details. This version has been accepted for publication by APDE

References in corpus (3)

• On classification of non-unital amenable simple C*-algebras, III, the range and the reduction
• Equivariant ${\mathcal{O}}_2$-absorption theorem for exact groups
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras