$C^*$-Algebras of Irreversible Dynamical Systems

http://dx.doi.org/10.4153/CJM-2006-003-x
Canad. J. Math. 58(2006), 39-63

  Published:2006-02-01
 Printed: Feb 2006

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.