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

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.

Minor typo errors corrected; conclusion section slightly revised. Comments are welcome!