$\mathcal{Z}$-stability for $\mathrm C^*$-algebras of minimal line-bundle-twisted homeomorphisms with the small boundary property
Marzieh Forough, Ja A Jeong, Karen R. Strung
TL;DR
This work extends the $\mathcal{Z}$-stability and Elliott-classification program to Cuntz–Pimsner algebras arising from minimal dynamics twisted by line bundles, by exploiting orbit-breaking subalgebras and recursive subhomogeneous decompositions. Under the small boundary property, the twisted algebras $\mathcal{O}_{C(X)}(\Gamma(\mathscr{V},\alpha))$ are $\mathcal{Z}$-stable, and their orbit-breaking subalgebras provide a structurally tractable path to this stability; the authors also show that tensor products of such algebras remain $\mathcal{Z}$-stable even without SBP, significantly broadening the class of mean-dimension-positive systems that are classifiable by the Elliott invariant. The methodological core combines SBP-based partition-of-unity constructions with RSH approximations to control dimension growth and radius of comparison, enabling robust stability results for a natural dynamical family of $\mathrm{C}^*$-algebras. The results yield concrete classification and regularity outcomes for a broad class of twisted dynamical C*-algebras and illuminate stability under tensor products in settings beyond mean-dimension-zero dynamics.
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.