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

Abstract

In this paper we study Cuntz--Pimsner algebras associated to ${\mathrm{C}}^{\ast }$-correspondences over commutative ${\mathrm{C}}^{\ast }$-algebras from the point of view of the ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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.

42 pages. Final version. To appear in Trans. Amer. Math. Soc