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)



Discrete groups actions and corresponding modules

Author: E. V. Troitsky
Journal: Proc. Amer. Math. Soc. 131 (2003), 3411-3422
MSC (2000): Primary 37Bxx, 46L08; Secondary 47B48
Published electronically: March 25, 2003
MathSciNet review: 1990630
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We address the problem of interrelations between the properties of an action of a discrete group $\Gamma$ on a compact Hausdorff space $X$ and the algebraic and analytical properties of the module of all continuous functions $C(X)$ over the algebra of invariant continuous functions $C_\Gamma(X)$. The present paper is a continuation of our joint paper with M. Frank and V. Manuilov. Here we prove some statements inverse to the ones obtained in that paper: we deduce properties of actions from properties of modules. In particular, it is proved that if for a uniformly continuous action the module $C(X)$ is finitely generated projective over $C_\Gamma (X)$, then the cardinality of orbits of the action is finite and fixed. Sufficient conditions for existence of natural conditional expectations $C(X)\to C_\Gamma(X)$ are obtained.

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

  • 1. E. Andruchow and D. Stojanoff, Geometry of conditional expectations of finite index, Internat. J. Math. 5 (1994), 169-178. MR 95e:46085
  • 2. M. Baillet, Y. Denizeau, and J.-F. Havet, Indice d'une esperance conditionelle, Compos. Math. 66 (1988), 199-236. MR 90e:46050
  • 3. N. Dunford and J. T. Schwartz, Linear operators. I. General theory, Interscience, New York, 1958. MR 22:8302
  • 4. M. Frank and E. Kirchberg, On conditional expectations of finite index, J. Oper. Theory 40 (1998), no. 1, 87-111. MR 2000k:46080
  • 5. M. Frank, V. M. Manuilov, and E. V. Troitsky, On conditional expectations arising from group actions, Zeitschr. Anal. Anwendungen 16 (1997), 831-850. MR 2000e:46089
  • 6. E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, Springer-Verlag, New York, 1963. MR 28:158
  • 7. P. Jolissaint, Indice d'esperances conditionnelles et algèbres de von Neumann finies, Math. Scand. 68 (1991), 221-246. MR 93g:46059
  • 8. V. Jones, Index of subfactors, Invent. Math. 41 (1981), 1-25. MR 84d:46097
  • 9. G. G. Kasparov, Hilbert C*-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), 133-150. MR 82b:46074
  • 10. M. Khoshkam, Hilbert C*-modules and conditional expectations on crossed products, J. Austral. Math. Soc. (Series A) 61 (1996), 106-118. MR 97i:46100
  • 11. E. C. Lance, Hilbert C*-modules - a toolkit for operator algebraists, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, England, 1995. MR 96k:46100
  • 12. V. M. Manuilov and E. V. Troitsky, Hilbert C*- and W*-modules and their morphisms, J. Math. Sci. (New York) 98 (2000), no. 2, 137-201. MR 2001k:46094
  • 13. A. S. Mishchenko and A. T. Fomenko, The index of elliptic operators over C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), 831-859, English translation, Math. USSR-Izv. 15, 87-112, 1980. MR 81i:46075
  • 14. M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Scient. Ec. Norm. Sup. 19 (1986), 57-106. MR 87m:46120
  • 15. M. Rieffel, Integrable and proper action on C*-algebras, and square-integrable representations of groups, E-print, 1998.
  • 16. W. Rinow, Lehrbuch der Topologie, Dt. Verlag Wiss., Berlin, 1975. MR 58:24157
  • 17. V. Seregin, Reflexivity of C*-Hilbert modules arising from group actions, Moscow Univ. Math. Bull. (2002), to appear.
  • 18. Y. Watatani, Index for C*-subalgebras, Memoirs Amer. Math. Soc., vol. 424, AMS, Providence, 1990. MR 90i:46104

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37Bxx, 46L08, 47B48

Retrieve articles in all journals with MSC (2000): 37Bxx, 46L08, 47B48

Additional Information

E. V. Troitsky
Affiliation: Department of Mechanics and Mathematics, Moscow State University, 119 899 Moscow, Russia

Keywords: Discrete groups, discrete noncommutative dynamical systems, Hilbert C*-modules
Received by editor(s): October 8, 2001
Received by editor(s) in revised form: May 21, 2002
Published electronically: March 25, 2003
Additional Notes: This work was partially supported by the RFBR (Grant 02-01-00572) and by the President of RF (Grant 00-15-99263)
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society