Group amenability and actions on 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 F₂, then there exist uncountably many, non-cocycle conjugate strongly outer actions of G on any tracial, unital, separable C*-algebra that absorbs tensorially the Jiang-Su algebra. Similar conclusions hold for outer actions on McDuff II₁ factors. We moreover show that G is amenable if and only if the Bernoulli shift on any finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods are inspired by Jones' work [27], and consist in a careful study of weak containment for the Koopman representations of certain generalized Bernoulli actions.

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