$\mathcal{Z}$-stability for $\mathrm C^*$-algebras of minimal line-bundle-twisted homeomorphisms with the small boundary property

TL;DR

This paper proves that certain Cuntz--Pimsner algebras associated with minimal line-bundle-twisted homeomorphisms are $ ext{Z}$-stable if the system has the small boundary property, leading to classification by the Elliott invariant.

Contribution

It establishes $ ext{Z}$-stability for these algebras under the small boundary property and shows the tensor product of such algebras is always $ ext{Z}$-stable, even without this property.

Findings

Cuntz--Pimsner algebras are $ ext{Z}$-stable with the small boundary property.

Tensor products of these algebras are always $ ext{Z}$-stable.

Applicable to systems with positive mean dimension.

Abstract

In this paper we show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twisted by line bundles, along with their orbit-breaking subalgebras, are $\mathcal{Z}$-stable whenever the underlying dynamical system has the small boundary property. This entails that this class is classified by the Elliott invariant. Furthermore, we show that the tensor product of two such $\mathrm{C}^*$-algebras is always $\mathcal{Z}$-stable, without assuming the small boundary property. In particular this applies to $\mathrm{C}^*$-algebras arising from systems with positive mean dimension.