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

Abstract

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^{\ast }$-algebra, and let $\alpha :G\to \mathrm{Aut}(A)$ 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 ${\partial }_{e}T(A)$ is furthermore compact, has finite covering dimension, and the orbit space ${\partial }_{e}T(A)/G$ 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.

51 pages; this preprint supersedes and expands arXiv:1811.00447

References in corpus (3)

• Completely positive maps of order zero
• Strongly self-absorbing C*-algebras are Z-stable
• Actions of amenable groups and crossed products of Z-absorbing C*-algebras