Spaces of Continuous and Measurable Functions Invariant under a Group Action

TL;DR

This paper characterizes spaces of continuous and measurable functions on compact spaces that remain invariant under group actions, extending classical results to broader contexts.

Contribution

It generalizes Nagel and Rudin's 1976 results to characterize invariant function spaces under transitive group actions on compact Hausdorff spaces.

Findings

Characterization of invariant function spaces under group actions

Extension of classical invariance results to new settings

Broader understanding of symmetry in function spaces

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 M\"obius invariant spaces of continuous and measurable functions defined on the unit sphere in $\mathbb{C}^n$.