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

Michael Frank ¹   ; Vladimir Manuilov ²   ; Evgenij Troitsky ³

¹ FB IMN HTWK Leipzig Postfach 301166 D-04251 Leipzig, Germany
² Department of Mechanics and Mathematics Moscow State University 119991 GSP-1 Moscow, Russia and Harbin Institute of Technology Harbin, P.R. China
³ Department of Mechanics and Mathematics Moscow State University 119991 GSP-1 Moscow, Russia

Studia Mathematica, Tome 200 (2010) no. 2, pp. 131-148

Résumé

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.

Détail

DOI : 10.4064/sm200-2-2

Keywords: continuous actions topological groups compact hausdorff spaces investigated which induce almost periodic functions corresponding commutative * algebra unique invariant mean group resulting averaging allows derive * valued inner product hilbert * module which serve environment describe characteristics group action lyapunov stable actions derived invariant mean phi continuous phi induced * valued inner product corresponds conditional expectation fixed point algebra action defined averaging orbits self duality hilbert * module orbits shown have cardinality stable actions compact metric spaces rise * reflexive hilbert * modules cardinality finite orbits uniformly bounded number closures infinite orbits finite number examples illustrate typical situations appearing beyond classified cases