The Modular Stone-von Neumann Theorem

TL;DR

This paper generalizes the Stone-von Neumann Theorem to modular representations of group actions on C*-algebras using nonabelian duality, extending previous results to a broader class of groups and representations.

Contribution

It introduces a new approach based on Hilbert C*-modules to prove a far-reaching generalization of the classical theorem, improving efficiency and applicability.

Findings

Generalization of Stone-von Neumann Theorem to modular group representations

New Hilbert C*-module result simplifies proofs and broadens scope

Extension of Covariant Stone-von Neumann Theorem for nonabelian groups

Abstract

In this paper, we use the tools of nonabelian duality to formulate and prove a far-reaching generalization of the Stone-von Neumann Theorem to modular representations of actions and coactions of locally compact groups on elementary $ C^{\ast} $-algebras. This greatly extends the Covariant Stone-von Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and the second author. Our approach is based on a new result about Hilbert $ C^{\ast} $-modules that is simple to state yet is widely applicable and can be used to streamline many previous arguments, so it represents an improvement -- in terms of both efficiency and generality -- in a long line of results in this area of mathematical physics that goes back to J. von Neumann's proof of the classical Stone-von Neumann Theorem.