Spaces of Continuous and Measurable Functions Invariant under a Group Action

Abstract

In this paper we characterize spaces of continuous and L p -functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes Nagel and Rudin's 1976 results concerning unitarily and Möbius invariant spaces of continuous and measurable functions defined on the unit sphere in C n .

Spaces of Bounded Measurable Functions Invariant under a Group Action

“…The following definition has appeared in several sources, such as [1,6,7], but no attribution is given. The last citation is the specific case of the unitary group acting on the unit sphere in C n .…”

“…Definition 3 (Definition 3.1 [1]). For each x ∈ X, the space H(x) is the set of all continuous functions that are unchanged by the action of any element of G which stabilizes x.…”

“…This collection is necessarily unique. Definition 5 (Definition 3.7 [1]). We define π i to be the projection of L 2 (µ) onto H i .…”

“…The motivation for this paper is to provide an analogue to the results of [1], in which it is shown that when the collection G exists, every uniformly closed space of continuous functions on X that is invariant under the group action can be decomposed into the closure of the direct sum of some subcollection of G . The same is also true of the L p -functions, with measure µ, for 1 ≤ p < ∞, where closure is taken in the usual norm-topology on L p (µ).…”

“…The case of L ∞ -functions with the norm-topology is not established in [1] because the map α → f • ϕ α , where ϕ α denotes the action of α ∈ G on X, from G into the L ∞ -functions need not be continuous under the norm-topology. However, when considering the space of L ∞ -functions with the weak*-topology, an analogous result can be established (Theorem 3), which states that when the same collection G of continuous functions on X from [1] exists (Definition 4), all weak*-closed invariant spaces of L ∞ -functions can be constructed by closing the direct sum of some subcollection of G in the weak*-topology.…”