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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
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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.

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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

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