On the Exel crossed product of topological covering maps

Abstract

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^{\ast }$-algebras $\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ into $\cros$ is a maximal abelian $C^{\ast }$-subalgebra of $\cros$; any nontrivial two sided ideal of $\cros$ has non-zero intersection with the imbedded copy of $C(X)$; a certain natural representation of $\cros$ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product $C^{\ast }$-algebras of homeomorphism dynamical systems.

References in corpus (2)

• Compactly supported wavelets and representations of the Cuntz relations
• C*-crossed products and shift spaces

Cited by in corpus (3)

• Topological aperiodicity for product systems over semigroups of Ore type
• Maximal commutative subrings and simplicity of Ore extensions
• Topological freeness for $C^{\ast }$-correspondences