Equivariant $\mathcal{Z}$-stability for single automorphisms on simple $C^{\ast }$-algebras with tractable trace simplices

Abstract

Let $A$ be an algebraically simple, separable, nuclear, $\mathcal{Z}$-stable $C^{\ast }$-algebra for which the trace space $T(A)$ is a Bauer simplex and the extremal boundary ${\partial }_{e}T(A)$ has finite covering dimension. We prove that each automorphism $\alpha$ on $A$ is cocycle conjugate to its tensor product with the trivial automorphism on the Jiang-Su algebra. At least for single automorphisms this generalizes a recent result by Gardella-Hirshberg-Vaccaro. If $\alpha$ is strongly outer as an action of $\mathbb{Z}$, we prove it has finite Rokhlin dimension with commuting towers. As a consequence it tensorially absorbs any automorphism on the Jiang-Su algebra.

38 pages, accepted version. Final version will appear in Mathematische Zeitschrift