Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Entropy for $ C^*$-algebras with tracial rank zero


Author: Rui Okayasu
Journal: Proc. Amer. Math. Soc. 138 (2010), 3609-3621
MSC (2010): Primary 46L55; Secondary 28D20, 46L40
DOI: https://doi.org/10.1090/S0002-9939-10-10475-4
Published electronically: May 7, 2010
MathSciNet review: 2661560
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Voiculescu's topological approximation entropy is extended to automorphisms on unital simple $ C^*$-algebras with tracial rank zero. Several expected properties are shown. We also consider the value of our entropy for a cat map on the non-commutative torus.


References [Enhancements On Off] (What's this?)

  • 1. Brown, N. P. Topological entropy in exact $ C\sp *$-algebras. Math. Ann. 314 (1999), no. 2, 347-367. MR 1697449 (2000g:46093)
  • 2. Blackadar, B. Comparison theory for simple $ C\sp *$-algebras. Operator algebras and applications, Vol. 1, pp. 21-54, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge, 1988. MR 996438 (90g:46078)
  • 3. Connes, A.; Narnhofer, H.; Thirring, W. Dynamical entropy of $ C^*$-algebras and von Neumann algebras. Comm. Math. Phys. 112 (1987), no. 4, 691-719. MR 910587 (89b:46078)
  • 4. Connes, A.; Størmer, E. Entropy for automorphisms of $ II_1$ von Neumann algebras. Acta Math. 134 (1975), no. 3-4, 289-306. MR 0454657 (56:12906)
  • 5. Elliott, G. A.; Evans, D. E. The structure of the irrational rotation $ C\sp *$-algebra. Ann. of Math. (2) 138 (1993), no. 3, 477-501. MR 1247990 (94j:46066)
  • 6. Elliott, G. A.; Lin, Q. Cut-down method in the inductive limit decomposition of non-commutative tori. J. London Math. Soc. (2) 54 (1996), no. 1, 121-134. MR 1395072 (98e:46067)
  • 7. Golodets, V. Y.; Neshveyev, S. V. Entropy of automorphisms of $ \textrm{II}_1$-factors arising from the dynamical systems theory. J. Funct. Anal. 181 (2001), no. 1, 14-28. MR 1818110 (2002i:46069)
  • 8. Hu, S.; Lin, H.; Xue, Y. The tracial topological rank of $ C^*$-algebras. II. Indiana Univ. Math. J. 53 (2004), no. 6, 1578-1603. MR 2106337 (2005i:46063)
  • 9. Lin, H. The tracial topological rank of $ C\sp *$-algebras. Proc. London Math. Soc. (3) 83 (2001), no. 1, 199-234. MR 1829565 (2002e:46063)
  • 10. Lin, H. An introduction to the classification of amenable $ C^*$-algebras. World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1884366 (2002k:46141)
  • 11. Lin, H.; Phillips, N. C. Crossed products by minimal homeomorphisms. Preprint, 2004, arXiv:math.OA/0408291.
  • 12. Neshveyev, S.; Størmer, E. Dynamical entropy in operator algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 50. Springer-Verlag, Berlin, 2006. MR 2251116 (2007m:46108)
  • 13. Rieffel, M. A. Projective modules over higher-dimensional noncommutative tori. Canad. J. Math. 40 (1988), no. 2, 257-338. MR 941652 (89m:46110)
  • 14. Voiculescu, D. Dynamical approximation entropies and topological entropy in operator algebras. Comm. Math. Phys. 170 (1995), no. 2, 249-281. MR 1334396 (97b:46082)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L55, 28D20, 46L40

Retrieve articles in all journals with MSC (2010): 46L55, 28D20, 46L40


Additional Information

Rui Okayasu
Affiliation: Department of Mathematics Education, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan
Email: rui@cc.osaka-kyoiku.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-10-10475-4
Keywords: $C^*$-algebras, entropy, trace
Received by editor(s): March 4, 2009
Received by editor(s) in revised form: December 1, 2009, and January 8, 2010
Published electronically: May 7, 2010
Additional Notes: The author was supported in part by the Japan Society for the Promotion of Science
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society