The Modular Stone-von Neumann Theorem
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.08997
- arXiv:
- arXiv:2109.08997
- Bibcode:
- 2021arXiv210908997H
- Keywords:
-
- Mathematics - Operator Algebras;
- Mathematical Physics;
- Mathematics - Dynamical Systems;
- Mathematics - Functional Analysis;
- Mathematics - Representation Theory;
- 46L55 (Primary) 22D25;
- 22D35;
- 43A65;
- 46L06;
- 81R15;
- 81S05 (Secondary)
- E-Print:
- 14 pages. Minor typo errors corrected and exposition much more streamlined. To appear in the Journal of Operator Theory