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 of random walks on groups and the Macaev norm


Author: D. Voiculescu
Journal: Proc. Amer. Math. Soc. 119 (1993), 971-977
MSC: Primary 47N30; Secondary 47B06, 47B10, 47D50, 60B15
DOI: https://doi.org/10.1090/S0002-9939-1993-1151816-4
MathSciNet review: 1151816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a discrete group on which there is a finitary random walk with positive entropy satisfies a certain condition involving the Macaev norm. This links the entropy of random walks on groups to the author's work on quasicentral approximate units relative to normed ideals in perturbation theory. On the other hand, the condition we are considering is also an analogue for the Macaev norm of Yamasaki's hyperbolicity condition.


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

  • [1] A. Avez, Entropie des groupes de type fini, C. R. Acad. Sci. Paris Sér. A 275 (1972), 1363-1366. MR 0324741 (48:3090)
  • [2] -, Théorème de Choquet-Deny pour les groupes à croissance non-exponentielle, C. R. Acad. Sci. Paris Sér. A 279 (1974), 25-28. MR 0353405 (50:5888)
  • [3] -, Harmonic functions on groups in differential geometry and relativity, Math. Phys. Appl. Math. 3 (1976), 27-32. MR 0507229 (58:22389)
  • [4] D. Bernier, Quasicentral approximate units for the discrete Heisenberg group, preprint, Berkeley, 1991. MR 1284881 (95h:47040)
  • [5] D. Bisch, Entropy of groups and subfactors, preprint, UCLA, 1990. MR 1144689 (93e:46076)
  • [6] A. Connes, Trace de Dixmier, modules de Fredholm et géométrie riemanniène, Nuclear Phys. B Proc. Suppl. 5 (1988), 65-70. MR 1002957 (90d:58014)
  • [7] Y. Derrienic, Quelques applications du théorème ergodique sous-additif, Astérisque 74 (1980), 183-201. MR 588163 (82e:60013)
  • [8] E. B. Dynkin and M. B. Malyutov, Random walks on groups with a finite number of generators, Soviet Math. Dokl. 2 (1961), 399-402.
  • [9] H. Furstenberg, Random walks on discrete subgroups of Lie groups, Adv. Probab. Related Topics, vol. 1, Dekker, New York, 1971, pp. 1-63. MR 0284569 (44:1794)
  • [10] I. C. Gohberg and M. G. Krein, Introduction to the theory of non-selfadjoint operators, Nauka, Moscow, 1965. MR 0220070 (36:3137)
  • [11] V. A. Khaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 3 (1983), 457-490. MR 704539 (85d:60024)
  • [12] V. A. Khaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, preprint.
  • [13] S. Popa, Sous-facteurs, actions de groupes et cohomologie, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 771-776. MR 1054961 (91i:46069)
  • [14] N. T. Varopoulos, Long range estimates for Markov chains, Bull. Sci. Math. (2) 109 (1985), 225-252. MR 822826 (87j:60100)
  • [15] D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators. I, J. Operator Theory 1 (1979), 3-37; II, J. Operator Theory 5 (1981), 77-100. MR 613049 (83f:47014)
  • [16] -, Hilbert space operators modulo normed ideals, Proc. Internat. Congr. Math., Warsaw, 1983, pp. 1041-1047. MR 804756 (87b:47049)
  • [17] -, On the existence of quasicentral approximate units relative to normed ideals, Part I, J. Funct. Anal. 91 (1990), 1-36. MR 1054113 (91m:46089)
  • [18] -, Entropy of dynamical systems and perturbations of operators. I, Ergodic Theory Dynamical Systems 11 (1991), 779-786; II, Houston Math. J. 17 (1991), 651-661. MR 1145622 (93b:46130)
  • [19] M. Yamasaki, Parabolic and hyperbolic infinite networks, Hiroshima Math. J. (1977), 135-146. MR 0429377 (55:2395)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47N30, 47B06, 47B10, 47D50, 60B15

Retrieve articles in all journals with MSC: 47N30, 47B06, 47B10, 47D50, 60B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1151816-4
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society