Spaces of Bounded Measurable Functions Invariant Under a Group Action

TL;DR

This paper characterizes invariant spaces of essentially bounded functions under group actions on compact spaces, extending previous work on specific invariances like unitarity and M"obius transformations.

Contribution

It generalizes earlier results by providing a comprehensive characterization of $L^ Infty$-spaces invariant under transitive, continuous group actions on compact Hausdorff spaces.

Findings

Provides a complete description of invariant $L^ Infty$-spaces under group actions.

Extends previous results from specific invariances to general group actions.

Establishes a framework applicable to various symmetry groups in functional analysis.

Abstract

In this paper we characterize spaces of $L^\infty$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes the author's 2021 results concerning the specific case of unitarily and M\"obius invariant spaces of $L^\infty$-functions defined on the unit sphere in $\mathbb{C}^n$.