Exel, R.; Vershik, A. \(C^*\)-algebras of irreversible dynamical systems. (English) Zbl 1104.46037 Can. J. Math. 58, No. 1, 39-63 (2006). Summary: 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. Cited in 34 Documents MSC: 46L55 Noncommutative dynamical systems 37A55 Dynamical systems and the theory of \(C^*\)-algebras Keywords:endomorphism crossed product; presentations; circle actions PDFBibTeX XMLCite \textit{R. Exel} and \textit{A. Vershik}, Can. J. Math. 58, No. 1, 39--63 (2006; Zbl 1104.46037) Full Text: DOI arXiv