An easy proof of the Stone-von Neumann-Mackey theorem

TL;DR

This paper provides a straightforward proof of the Stone-von Neumann-Mackey theorem for Heisenberg groups linked to locally compact abelian groups, utilizing the Peter-Weyl theorem and Fourier analysis.

Contribution

It offers a simplified proof of the theorem by leveraging classical harmonic analysis tools, assuming known structural results of locally compact abelian groups.

Findings

Proof simplifies understanding of the theorem

Utilizes Peter-Weyl theorem and Fourier transforms

Assumes known structure of locally compact abelian groups

Abstract

The Stone-von Neumann-Mackey Theorem for Heisenberg groups associated to locally compact abelian groups is proved using the Peter-Weyl theorem and the theory of Fourier transforms for finite dimensional real vector spaces. A theorem of Pontryagin and van Kampen on the structure of locally compact abelian groups (which is evident in any particular case) is assumed.