The C*-algebras associated with minimal homeomorphisms of the cantor set

Abstract

We investigate the structure of the C*-algebras associated with minimal homeomorphisms of the Cantor set via the crossed product construction. These C*-algebras exhibit many of the same properties as approximately finite dimensional (or AF) C*-algebras. Specifically, each non-empty closed subset of the Cantor set is shown to give rise, in a natural way, to an AF-subalgebra of the crossed product and we analyze these subalgebras. Results of Versik show that the crossed product may be embedded into an AF-algebra. We show that this embedding induces an order isomorphism at the level of Ko-groups. We examine examples arising from the theory of interval exchange transformations. © 1989 by Pacific Journal of Mathematics.