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 $\phi \in C(X)$, 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.