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# $C^*$-Algebras of Irreversible Dynamical Systems

Published:2006-02-01
Printed: Feb 2006
• R. Exel
• A. Vershik
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## Abstract

We show that certain $C^*$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.
 MSC Classifications: 46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20] 37A55 - Relations with the theory of $C^*$-algebras [See mainly 46L55]

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