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.