Discrete groups actions and corresponding modules

Abstract

Abstract. We address the problem of interrelations between the properties of an action of a discrete group Γ 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 Γ (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 Γ (X), then the cardinality of orbits of the action is finite and fixed. Sufficient conditions for existence of natural conditional expectations C(X) → C Γ (X) are obtained.

Quantization of branched coverings

“…Recall (cf. [24]) that a unital C * -algebra is said to be MI (module infinite) if each countably generated Hilbert module over it is projective finitely generated if and only if it is self-dual. The C * -algebra C(X) of this example is MI by [24,Theorem 33], therefore C(Y ) is not a self-dual module over it.…”

“…92-93] (see also Theorem 5.7 below). To prove the inverse implication let us remark that X does not have isolated points because it is connected, so the C * -algebra C(X) is MI by [24,Theorem 33]. According to our assumptions and Lemma 4.1 the C(X)-module C(Y ) has to be both countably generated and self-dual.…”

“…Acknowledgement: The authors are grateful to M. Frank, V. M. Manuilov, A. S. Mishchenko and T. Schick for helpful discussions. In particular, A. S. Mishchenko (after a talk of one of us (ET) on the results of [24]) has posed some of the problems solved in the present paper.…”

“…Our research continues the research on spaces and modules arising from discrete group actions (cf. [9,24]). Apart from the mentioned above papers let us indicate the research of Buchstaber, Rees, Gugnin and others on algebraic definition and topological applications of Dold-Smith ramified coverings (see, e.g., [3,4,12]).…”

Hilbert C*-modules from group actions: beyond the finite orbits case

“…Investigating continuous group actions on topological spaces several mathematical approaches may be applied. In the present paper the authors continue their work started in [5,18] which relies on the Gel'fand duality of locally compact Hausdorff spaces and commutative C * -algebras. In the dual picture some well-known results from functional analysis and noncommutative geometry can be applied to get new insights, often also for related noncommutative situations of group actions on general C * -algebras.…”