𝐶*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

Abstract

In this paper we study Cuntz–Pimsner algebras associated to 𝐶*-correspondences over commutative C ∗ \mathrm {C}^* -algebras from the point of view of the C ∗ \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 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 C ∗ \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 X twisted by a line bundle over X X , we introduce orbit-breaking subalgebras. With no assumptions on the dimension of X X , we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X X is finite, they are furthermore Z \mathcal {Z} -stable and hence classified by the Elliott invariant.