Discrete groups actions and corresponding modules

Troitsky, E.

doi:10.1090/S0002-9939-03-07043-6

Discrete groups actions and corresponding modules
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Proc. Amer. Math. Soc. 131 (2003), 3411-3422 Request permission

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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