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

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 is an element of 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.