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arXiv · 2105.04469

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

Lise Wouters
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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.

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Lise Wouters. 2021-05-10. Equivariant $\mathcal{Z}$-stability for single automorphisms on simple $C^*$-algebras with tractable trace simplices. https://arxiv.org/abs/2105.04469

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