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Strongly outer actions of amenable groups on Z-stable C^*-algebras · arXivDesk

1811.00447Nov 1, 2018Version 3: 47 pages. Deep revision of the previous version. In addition to fixing several gaps, we improved the results on Rokhlin dimension: we are now able to handle arbitrary residually finite amenable groups

Strongly outer actions of amenable groups on $\mathcal{Z}$ -stable $C^*$ -algebras

Eusebio Gardella, Ilan Hirshberg

Operator Algebras Functional Analysis

Abstract

Let $A$ be a separable, unital, simple, $\mathcal{Z}$ -stable, nuclear $C^*$ -algebra, and let $α\colon G\to \mathrm{Aut}(A)$

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be an action of a countable amenable group

G

. If the trace space

T(A)

is a Bauer simplex and the action of

G

on

\partial_eT(A)

has finite orbits and Hausdorff orbit space, we show that

α

is strongly outer if and only if

α\otimes\mathrm{id}_{\mathcal{Z}}

has the weak tracial Rokhlin property. If

G

is moreover residually finite, then these conditions are also equivalent to

α\otimes\mathrm{id}_{\mathcal{Z}}

having finite Rokhlin dimension (in fact, at most 2). When the covering dimension of

\partial_eT(A)

is finite, we prove that

α

is cocycle conjugate to

α\otimes\mathrm{id}_{\mathcal{Z}}

. In particular, the equivalences above hold for

α

in place of

α\otimes\mathrm{id}_{\mathcal{Z}}

.

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