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${{C}^{*}}$-Algebras of Irreversible Dynamical Systems

Published online by Cambridge University Press:  20 November 2018

R. Exel  and
A. Vershik
R. Exel
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, Brazil e-mail: exel@mtm.ufsc.br
A. Vershik
Affiliation:
Russian Academy of Sciences, St. Petersburg Russia e-mail: vershik@pdmi.ras.ru
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Abstract

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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.

Keywords

46L5537A55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 1 , 01 February 2006 , pp. 39 - 63
Copyright
Copyright © Canadian Mathematical Society 2006

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