Certain invariant spaces of bounded measurable functions on a sphere

Spaces of Bounded Measurable Functions Invariant under a Group Action

“…In this section, we prove Lemma 1, which we note is an analogue to Lemma 4.4 of [1] and a generalization of Lemma 4.2 from [2]. The proof requires the following lemmas:…”

“…This paper also serves as a generalization of a result in [2], in which the author establishes the particular case for the L ∞ -functions defined on the unit sphere of C n acted on by the unitary group. This in turn was motivated by the work of Nagel and Rudin in [3], in which it is shown that there exists a collection C of (minimal and invariant) spaces of continuous functions on the unit sphere of C n such that each closed unitarily invariant space of continuous or L p -functions on the sphere decomposes into the closure of the direct sum of some subcollection of C .…”