# The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on $ C^{\ast} $-Algebras of Compact Operators

## TL;DR

This paper extends the Stone-von Neumann Theorem to $ C^{st} $-dynamical systems involving abelian groups and compact operators, using Hilbert $ C^{st} $-modules and introducing new commutation relations.

## Contribution

It introduces a covariant version of the Stone-von Neumann Theorem for abelian group actions on $ C^{st} $-algebras of compact operators, utilizing Hilbert $ C^{st} $-modules.

## Key findings

- Proves a uniqueness theorem for $ C^{st} $-dynamical systems with abelian groups.
- Develops a representation of Weyl relations on Hilbert $ C^{st} $-modules.
- Simplifies the proof of Takai-Takesaki Duality using new Hilbert $ C^{st} $-module results.

## Abstract

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),\alpha) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert $ C^{\ast} $-modules to significantly shorten the length of Iain Raeburn's well-known proof of Takai-Takesaki Duality.