Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean is continuous on X for any ϕ ∈ C(X), and the induced C*-valued inner product corresponds to a conditional expectation from C(X) onto the fixed-point algebra of the action defined by averaging on orbits. In the case of self-duality of the Hilbert C*-module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C*-reflexive Hilbert C*-modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-2,
author = {Michael Frank and Vladimir Manuilov and Evgenij Troitsky},
title = {Hilbert C*-modules from group actions: beyond the finite orbits case},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {131-148},
zbl = {1210.46044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-2}
}
Michael Frank; Vladimir Manuilov; Evgenij Troitsky. Hilbert C*-modules from group actions: beyond the finite orbits case. Studia Mathematica, Tome 196 (2010) pp. 131-148. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-2-2/