Hilbert C^*-modules from group actions: beyond the finite orbits 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 (φ x ) is continuous on X for any element 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 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.

Geometric essence of "compact" operators on Hilbert $C^*$-modules

“…Details and proofs can be found in books [7,11] and survey paper [10]. Some other directions joining Hilbert C * -modules and operator theory can be found in [3,14,1,15]. Definition 1.3.…”

On subspaces of invariant vectors

“…Remark 10. The original definition in [5] was different: for any x ∈ K and ǫ > 0 there must exist δ = δ(x, ε) > 0 such that d(gx, gy) < ε, for all g ∈ G, whenever d(x, y) < δ.…”

“…Lyapunov stable actions were introduced in [5]. It was shown in [5] that if the action is Lyapunov stable, then there is a conditional expectation on the subspace (actually, subalgebra) of invariant functions. In Theorem 15 we give a new proof of that.…”

“…Though for such representations the decomposition X = X π ⊕ Ann(X * π ) need not hold in general, it is shown that it holds if the action is nice, namely Lyapunov stable (Theorem 15). Lyapunov stable actions were introduced in [5]. It was shown in [5] that if the action is Lyapunov stable, then there is a conditional expectation on the subspace (actually, subalgebra) of invariant functions.…”

“…Moreover we construct a conditional expectation commuting with the representation and show that such an expectation is unique. Along the way we give a short proof of an assertion in [5] on the uniqueness of invariant measures. In section 4 we introduce lower semi-continuous actions, which is a wider class of actions than Lyapunov stable ones.…”