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 C(X)a < S (alpha,a"')a"center dot 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 embedding of C(X) into C(X)a < S (alpha,a"')a"center dot is a maximal abelian C (*)-subalgebra of C(X)a < S (alpha,a"')a"center dot; any nontrivial two sided ideal of C(X)a < S (alpha,a"')a"center dot has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)a < S (alpha,a"')a"center dot is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C (*)-algebras of homeomorphism dynamical systems.