A Note on the Peter-Weyl Theorem

TL;DR

This paper generalizes the Peter-Weyl Theorem by exploring functions on locally compact groups with large compact open subgroups, using classical representation theory concepts to approximate functions with well-known representative functions.

Contribution

It introduces a new generalization of the Peter-Weyl Theorem applicable to functions on certain locally compact groups with large compact open subgroups.

Findings

Functions on these groups can be approximated by functions locally identical to classical representative functions.

The approach extends classical representation theory concepts to broader classes of groups.

Provides a framework for analyzing functions on groups with large compact open subgroups.

Abstract

We introduce some classical concepts in the representation theory of compact groups, in order to use them for a new generalization of the Peter-Weyl Theorem. We mostly deal with functions on locally compact groups possessing large nontrivial compact open subgroups: in fact, we show that these functions can be approximated via others which are locally identical to the well known representative functions.