Journal of Operator Theory

Volume 89, Issue 2, Spring 2023 pp. 571-586.

The modular Stone-von Neumann theorem

Authors: Lucas Hall (1), Leonard Huang (2), John Quigg (3)
Author institution: (1) School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, U.S.A.
(2) Department of Mathematics and Statistics, University of Nevada, Reno, Reno, Nevada 89557, U.S.A.
(3) School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, U.S.A.

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2021sep18.2361
Keywords: crossed product, action, coaction, $ C^{\ast} $-correspondence, Morita equivalence, nonabelian duality, Stone-von Neumann theorem

Contents Full-Text PDF