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

Volume 200 / 2010

Michael Frank, Vladimir Manuilov, Evgenij Troitsky Studia Mathematica 200 (2010), 131-148 MSC: Primary 46L08; Secondary 43A60, 54H20. DOI: 10.4064/sm200-2-2

Abstract

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 $M(\phi _x)$ is continuous on $X$ for any $\phi \in C(X)$, 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.

Authors

  • Michael FrankFB IMN
    HTWK Leipzig
    Postfach 301166
    D-04251 Leipzig, Germany
    e-mail
  • Vladimir ManuilovDepartment of Mechanics and Mathematics
    Moscow State University
    119991 GSP-1 Moscow, Russia
    and
    Harbin Institute of Technology
    Harbin, P.R. China
    e-mail
  • Evgenij TroitskyDepartment of Mechanics and Mathematics
    Moscow State University
    119991 GSP-1 Moscow, Russia
    e-mail

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