C*-ALGEBRAS ASSOCIATED TO HOMEOMORPHISMS TWISTED BY VECTOR BUNDLES OVER FINITE DIMENSIONAL SPACES

In this paper we study Cuntz-Pimsner algebras associated to C*-correspondences over commutative C*-algebras from the point of view of the 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 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 Z-stable and hence classified by the Elliott invariant.