Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear $C^*$-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 discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$ are finite and that their cardinality is bounded. 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, these conditions are also equivalent to $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). If $\partial_eT(A)$ is furthermore compact, has finite covering dimension, and the orbit space $\partial_eT(A)/G$ is Hausdorff, we generalize results by Matui and Sato to show that $\alpha$ is cocycle conjugate to $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$, even if $\alpha$ is not strongly outer. In particular, in this case the equivalences above hold for $\alpha$ in place of $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property $\Gamma$ as technical tools of independent interest.