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

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.