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

TL;DR

This paper proves that automorphisms on certain simple, nuclear, $ ext{Z}$-stable $C^*$-algebras with tractable trace simplices are equivariantly $ ext{Z}$-stable, extending recent results and analyzing their Rokhlin dimension.

Contribution

It establishes equivariant $ ext{Z}$-stability for automorphisms on a broad class of $C^*$-algebras with finite-dimensional trace boundaries, generalizing prior work.

Findings

Automorphisms are cocycle conjugate to their tensor product with the trivial automorphism on $ ext{Z}$.

Strongly outer automorphisms have finite Rokhlin dimension with commuting towers.

Automorphisms tensorially absorb any automorphism on the Jiang-Su algebra.

Abstract

Let $A$ be an algebraically simple, separable, nuclear, $\mathcal{Z}$-stable $C^*$-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.