Dynamical comparison and $\mathcal{Z}$-stability for crossed products of simple $C^*$-algebras

TL;DR

This paper proves $\\mathcal{Z}$-stability for crossed products of certain $C^*$-algebras under group actions, using a new dynamical comparison technique and establishing automatic McDuffness in many cases.

Contribution

It introduces a weak dynamical comparison method and shows McDuffness is automatic in broad scenarios, leading to $\\mathcal{Z}$-stability results for crossed products.

Findings

Established $\mathcal{Z}$-stability for crossed products under mild assumptions.

Verified dynamical comparison in general settings.

Proved McDuffness with respect to invariant traces is automatic in many cases.

Abstract

We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a Bauer simplex with finite dimensional boundary $\partial_e T(A)$, and such that the induced action $G\curvearrowright \partial_eT(A)$ is free. If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension.