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

2023/12/23 by Hall, Lucas, Huang, Leonard, Krajczok, Jacek +1
#46L55 (Primary) #47L55 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2312.15264

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, α) 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.

Related

• Locally compact quantum groups in the von Neumann algebraic setting
• Amenability and covariant injectivity of locally compact quantum groups
• The Modular Stone-von Neumann Theorem
• The unitary implementation of a locally compact quantum group action
• Separation properties for positive-definite functions on locally compact quantum groups and for associated von Neumann algebras