Actions of nonamenable groups on $\mathcal{Z}$-stable $C^*$-algebras

TL;DR

This paper investigates how nonamenable groups act on $ ext{Z}$-stable C*-algebras, revealing that such actions are highly non-unique, especially for groups containing free subgroups, contrasting with the amenable case.

Contribution

It demonstrates that nonamenable groups have uncountably many non-cocycle conjugate strongly outer actions on Jiang-Su stable C*-algebras, and characterizes amenability via Bernoulli shifts.

Findings

Uncountably many non-cocycle conjugate actions for groups with free subgroups

Uniqueness of actions fails for all nonamenable groups

Amenability characterized by Bernoulli shift absorption

Abstract

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann algebras. We moreover show that $G$ is amenable if and only if the Bernoulli shift on a finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods consist in a careful study of weak containments of the Koopman representations of different Bernoulli-type actions.