The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups

Abstract

The Stone-von Neumann Theorem is a fundamental result which unified the competing quantum mechanical models of matrix mechanics and wave mechanics. It's mechanism of proof ultimately involved the study of unitary group representations on a Hilbert space. In this article, we continue the broad generalization set out in arxiv:1903.09351 and arxiv:2109.08997, analyzing representations of locally compact quantum dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models which appear in the literature, and for certain coactions (G, A, {\alpha}) recover the multiplicity results of arxiv:2109.08997. As a corollary, we develop a new criterion for identifying strongly regular locally compact quantum groups, related to the study of their dynamics on elementary C*-algebras.