A Criterion for $\mathcal{Z}$-Stability with Applications to Crossed Products

TL;DR

This paper establishes a criterion for $ ext{Z}$-stability in crossed products of C*-algebras, showing that under certain conditions, the crossed product retains $ ext{Z}$-stability, leading to classification results for nuclear purely infinite cases.

Contribution

It provides a new criterion for $ ext{Z}$-stability in crossed products of C*-algebras, extending the understanding of their structural properties.

Findings

Crossed product of $C(X,A)$ by a minimal automorphism is $ ext{Z}$-stable if $A$ is $ ext{Z}$-stable.

If $A$ is nuclear and purely infinite, the crossed product is a Kirchberg algebra.

The result builds on an argument by Toms and Winter to connect automorphism dynamics with algebraic stability.

Abstract

Building on an argument by Toms and Winter, we show that if $A$ is a simple, separable, unital, $\mathcal{Z}$-stable C*-algebra, then the crossed product of $C(X,A)$ by an automorphism is also Z-stable, provided that the automorphism induces a minimal homeomorphism on $X$. As a consequence, we observe that if $A$ is nuclear and purely infinite then the crossed product is a Kirchberg algebra.