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

TL;DR

This paper explores how group actions on compact spaces induce Hilbert C*-modules, revealing new structural insights especially for stable actions and those with bounded orbit sizes, extending beyond finite orbit cases.

Contribution

It introduces a framework linking group actions to Hilbert C*-modules via invariant means, extending analysis beyond finite and simple orbit structures.

Findings

Invariant mean induces a C*-valued inner product on the algebra.

Stable actions produce C*-reflexive Hilbert C*-modules.

All orbits have the same cardinality in the selfdual case.

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 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 uniformly continuous, Lyapunov stable actions the derived invariant mean $M(\phi_x)$ is continuous on $X$ for any element $\phi \in C(X)$, 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 selfduality 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.