On the Exel crossed product of topological covering maps

TL;DR

This paper explores the properties of Exel crossed product $C^*$-algebras associated with topological covering maps, establishing equivalences among topological freeness, maximal abelian subalgebras, ideal intersections, and faithfulness of representations.

Contribution

It generalizes known results for homeomorphism systems to non-invertible covering map dynamical systems, linking key algebraic and dynamical properties.

Findings

Topological freeness is equivalent to maximal abelian subalgebra property.

Nontrivial ideals intersect the embedded $C(X)$ nontrivially.

A natural representation of the crossed product is faithful under these conditions.

Abstract

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-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 $C^*$-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 $C^*$-algebras of homeomorphism dynamical systems.