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

Abstract

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $\alpha\colon G\to \mathrm{Aut}(A)$ 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 $\alpha$ is strongly outer if and only if $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, then these conditions are also equivalent to $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). When the covering dimension of $\partial_eT(A)$ is finite, we prove that $\alpha$ is cocycle conjugate to $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$. In particular, the equivalences above hold for $\alpha$ in place of $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$.