Search: MSC category 37A55 ( Relations with the theory of $C^$-algebras [See mainly 46L55] *$-algebras [See mainly 46L55] * )  Expand all Collapse all Results 1 - 3 of 3 1. CJM 2008 (vol 60 pp. 189) Lin, Huaxin  Furstenberg Transformations and Approximate Conjugacy Let$\alpha$and$\beta$be two Furstenberg transformations on$2$-torus associated with irrational numbers$\theta_1,\theta_2,$integers$d_1, d_2$and Lipschitz functions$f_1$and$f_2$. It is shown that$\alpha$and$\beta$are approximately conjugate in a measure theoretical sense if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z.$Closely related to the classification of simple amenable \CAs, it is shown that$\af$and$\bt$are approximately$K$-conjugate if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z$and$|d_1|=|d_2|.$This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35 2. CJM 2007 (vol 59 pp. 596) Itzá-Ortiz, Benjamín A.  Eigenvalues,$K$-theory and Minimal Flows Let$(Y,T)$be a minimal suspension flow built over a dynamical system$(X,S)$and with (strictly positive, continuous) ceiling function$f\colon X\to\R$. We show that the eigenvalues of$(Y,T)$are contained in the range of a trace on the$K_0$-group of$(X,S)$. Moreover, a trace gives an order isomorphism of a subgroup of$K_0(\cprod{C(X)}{S})$with the group of eigenvalues of$(Y,T)$. Using this result, we relate the values of$t$for which the time-$t$map on the minimal suspension flow is minimal with the$K$-theory of the base of this suspension. Categories:37A55, 37B05 3. CJM 2006 (vol 58 pp. 39) Exel, R.; Vershik, A. $C^*$-Algebras of Irreversible Dynamical Systems 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. Categories:46L55, 37A55