# 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.

## Key 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.