×

\(C^*\)-algebras of irreversible dynamical systems. (English) Zbl 1104.46037

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.

MSC:

46L55 Noncommutative dynamical systems
37A55 Dynamical systems and the theory of \(C^*\)-algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv