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On the Exel crossed product of topological covering maps

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.