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On the Exel Crossed Product of Topological Covering Maps

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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 C(X)⋊ α,ℒℕ 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)⋊ α,ℒℕ is a maximal abelian C *-subalgebra of C(X)⋊ α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C *-algebras of homeomorphism dynamical systems.

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Author notes

  1. Toke Meier Carlsen

    Present address: Department of Mathematical Sciences, The Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

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  1. Department of Mathematics & Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Toke Meier Carlsen

  2. Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden

    Sergei Silvestrov

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  1. Toke Meier Carlsen
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  2. Sergei Silvestrov
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Correspondence to Toke Meier Carlsen.

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Carlsen, T.M., Silvestrov, S. On the Exel Crossed Product of Topological Covering Maps. Acta Appl Math 108, 573–583 (2009). https://doi.org/10.1007/s10440-008-9372-6

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