$\mathrm{C}^*$-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

TL;DR

This paper investigates Cuntz--Pimsner algebras derived from aperiodic homeomorphisms twisted by vector bundles over finite-dimensional spaces, establishing their classification and structural properties within the C*-algebra framework.

Contribution

It demonstrates finite nuclear dimension and classification results for these algebras, introducing orbit-breaking subalgebras and analyzing their properties based on the dimension and rank of the vector bundle.

Findings

Cuntz--Pimsner algebras have finite nuclear dimension under certain conditions.

Classification by Elliott invariant for minimal homeomorphisms with finite-dimensional spaces.

Orbit-breaking subalgebras are simple, stably finite, and $ ext{Z}$-stable when the space dimension is finite.

Abstract

In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite-dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz--Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm{C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz--Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz--Pimsner algebra of a minimal homeomorphism of an infinite compact metric space $X$ twisted by a line bundle over $X$, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of $X$, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of $X$ is finite, they are furthermore $\mathcal{Z}$-stable and hence classified by the Elliott invariant.