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

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Frank M. , Manuilov V. , Troitsky E.

tom 200

nr 2

131-148

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 $M(ϕ_{x})$ 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.