Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T14:37:47.413Z Has data issue: false hasContentIssue false
  • English
  • Français

${{C}^{*}}$-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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Adji, S., Laca, M., Nilsen, M., and Raeburn, I., Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups. Proc. Amer.Math. Soc. 122(1994), 1133–1141.Google Scholar
[2] Archbold, R. J. and Spielberg, J. S., Topologically free actions and ideals in discrete C*-dynamical systems. Proc. EdinburghMath. Soc. 37(1994), 119–124.Google Scholar
[3] Arzumanian, V. A. and Vershik, A. M., Factor representations of the crossed product of a commutative C*-algebra with its endomorphism semigroup. Dokl. Akad. Nauk. SSSR 238(1978), 513–516 (Russian), Soviet Math. Dokl. 19(1978), 48–52 (English).Google Scholar
[4] Arzumanian, V. A. and Vershik, A. M., Star algebras associated with endomorphisms. In: Operator Algebras and Group Representations, Monogr. Stud. Math. 17, Pitman, Boston, MA, 1984, pp. 17–27.Google Scholar
[5] Bourbaki, N., Éléments de mathématique. XXV. Première partie. Livre VI: Intégration. Chapitre 6: Intégration vectorielle. Hermann, Paris, 1959.Google Scholar
[6] Boyd, S., Keswani, N., and Raeburn, I., Faithful representations of crossed products by endomorphisms. Proc. Amer. Math. Soc. 118(1993), 427–436.Google Scholar
[7] Bratteli, O., Inductive limits of finite dimensional C*-algebras. Trans. Amer. Math. Soc. 171(1972), 195–234.Google Scholar
[8] Deaconu, V., Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. 347(1995), 1779–1786.Google Scholar
[9] Effros, E. G. and Hahn, F., Locally Compact Transformation Groups and C*-Algebras. Mem. Amer. Math. Soc. 75, American Mathematical Society, Providence, RI, 1967.Google Scholar
[10] Exel, R., Circle actions on C*-algebras, partial automorphisms and a generalized Pimsner–Voiculescu exact sequence. J. Funct. Anal. 122(1994), 361–401. [arXiv:funct-an/9211001].Google Scholar
[11] Exel, R., A new look at the crossed product of a C*-algebra by an endomorphism. Ergodic Theory Dynam. Systems 23(2003), 1733–1750. [arXiv:math.OA/0012084].Google Scholar
[12] Exel, R., Crossed products by finite index endomorphisms and KMS states. J. Funct. Anal. 199(2003), 153–158.Google Scholar
[13] Exel, R., KMS states for generalized gauge actions on Cuntz-Krieger algebras (an application of the Ruelle-Perron-Frobenius Theorem). Bol. Soc. Brasil. Mat. (N.S.) 35(2004), no. 1, 1–12.Google Scholar
[14] Exel, R., Laca, M., and Quigg, J., Partial dynamical systems and C*-algebras generated by partial isometries. J. Operator Theory 47(2002), 169–186. [arXiv:funct-an/9712007].Google Scholar
[15] an Huef, A. and Raeburn, I., The ideal structure of Cuntz-Krieger algebras. Ergodic Theory Dynam. Systems 17(1997), 611–624.Google Scholar
[16] Laca, M. and Raeburn, I., Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139(1996), 415–440.Google Scholar
[17] Murphy, G. J., Crossed products of C*-algebras by endomorphisms. Integral Equations Operator Theory 24(1996), 298–319.Google Scholar
[18] Murray, F. J. and von Neumann, J., On rings of operators. Ann. of Math. 37(1936), 116–229.Google Scholar
[19] Pedersen, G. K., C*-Algebras and Their Automorphism Groups. London Mathematical Society Monographs 14, Academic Press, London, 1979,Google Scholar
[20] Power, S. C., Simplicity of C*-algebras of minimal dynamical systems. J. London Math. Soc. (2) 18(1978), 534–538.Google Scholar
[21] Renault, J., A Groupoid Approach to C*-Algebras. Lecture Notes in Mathematics 793, Springer, Berlin, 1980.Google Scholar
[22] Renault, J., Cuntz-like algebras. In: Operator Theoretical Methods, The Theta Foundation, Bucharest, 2000, pp. 371–386.Google Scholar
[23] Stacey, P. J., Crossed products of C*-algebras by* -endomorphisms. J. Aust. Math. Soc. Ser. A 54(1993), 204–212.Google Scholar
[24] Takesaki, M., Theory of Operator Algebras. I. Springer-Verlag, New York, 1979.Google Scholar
[25] Tomiyama, J., The Interplay between Topological Dynamics and Theory of C*-algebras. Lecture Notes Series 2, Research Institute of Mathematics, Seoul National University, 1992 Google Scholar
[26] Watatani, Y., Index for C* -Subalgebras. Mem. Amer. Math. Soc. 424, American Mathematical Society, Providence, RI, 1990.Google Scholar