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

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.