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A Note on the Peter-Weyl Theorem

Y. Bavuma, E. Stevenson, F. G. Russo

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.

A Note on the Peter-Weyl Theorem

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.
Paper Structure (3 sections, 5 theorems, 29 equations, 4 figures)

This paper contains 3 sections, 5 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General notions of topological algebra
  3. What happens to locally compact groups with compact open subgroups ?

Key Result

Lemma 2.5

Let $G$ be a locally compact group. Then there exists a map $\lambda:C_c(G,\mathbb{K})\rightarrow\mathbb{K}$ which is a Haar measure. Moreover, fix $\eta\in C_c^+(G,\mathbb{K})$ where $\eta\neq0$, and suppose $\mu:C_c(G,\mathbb{K})\rightarrow\mathbb{K}$ is another Haar measure. Then, for all $\varph Therefore there exists an $r \in ]0,+\infty[$ such that $\lambda=r\mu$.

Figures (4)

  • Figure 1: Demonstration of how a representative function may be lifted. Note that $G$ and $\mathbb{K}$ have been drawn like straight lines to give intuition, but will not in general have this structure.
  • Figure 2: Demonstration of how a lifted representative function can be obtained using multiple representative functions. Note that $G$ and $\mathbb{K}$ have been drawn like straight lines to give intuition, but will not in general have this structure.
  • Figure 3: Demonstration of how the function is split into various sections-- Note that $G$ and $\mathbb{K}$ have been drawn like straight lines to give intuition, but will not in general have this structure
  • Figure 4: Illustration of the algorithms where the approximating lifted representative functions are constructed. Note that $G$ and $\mathbb{K}$ have been drawn like straight lines to give intuition, but will not in general have this structure.

Theorems & Definitions (21)

  • Definition 2.1: $G$-Module, See hofmann, Definition 2.1 (i)
  • Definition 2.2: Strong Operator Topology and Representations, See hofmann, Chapter 2
  • Example 2.3
  • Definition 2.4: Support, $C_c(G,\mathbb{K})$ and $C_c^+(G,\mathbb{K})$, See stroppel, Definition 12.1
  • Lemma 2.5: Haar Measures for Locally Compact Groups, See stroppel, Theorems 12.20, 12.23 and 14.3
  • Definition 2.6: Haar Measure Notation, See stroppel Theorem 12.24
  • Definition 2.7: Normalized Haar Measure, See hofmann, Definition 2.6
  • Proposition 2.8: Existence and Uniqueness of Normalized Haar Measures, See hofmann, Theorem 2.8
  • Remark 2.9
  • Definition 2.10: Almost Invariant Elements, See hofmann, Definition 3.1
  • ...and 11 more