Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Entropy for -algebras with tracial rank zero

Author(s): Rui Okayasu
Journal: Proc. Amer. Math. Soc. 138 (2010), 3609-3621.
MSC (2010): Primary 46L55; Secondary 28D20, 46L40
Posted: May 7, 2010
MathSciNet review: 2661560
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Voiculescu's topological approximation entropy is extended to automorphisms on unital simple -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:

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 -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 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 -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 -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: 10.1090/S0002-9939-10-10475-4
PII: S 0002-9939(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
Posted: May 7, 2010
Additional Notes: The author was supported in part by the Japan Society for the Promotion of Science
Communicated by: Marius Junge
Copyright of article: Copyright 2010, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|