Vol. 282, No. 1, 2016

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Crossed product algebras and direct integral decomposition for Lie supergroups

Karl-Hermann Neeb and Hadi Salmasian

Vol. 282 (2016), No. 1, 213–232
Abstract

For every finite dimensional Lie supergroup (G,g), we define a C -algebra A := A(G,g) and show that there exists a canonical bijective correspondence between unitary representations of (G,g) and nondegenerate -representations of A. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations previously studied by Neeb, Salmasian, and Zellner.

For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated C -algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.

Keywords
crossed product algebras, unitary representations, Lie supergroups, Harish-Chandra pairs, direct integral decomposition, CCR algebras
Mathematical Subject Classification 2010
Primary: 17B15, 22E45, 47L65
Milestones
Received: 4 June 2015
Revised: 30 July 2015
Accepted: 27 August 2015
Published: 24 February 2016
Authors
Karl-Hermann Neeb
Department Mathematik
FAU Erlangen-Nürnberg
Cauerstr. 11
D-91058 Erlangen
Germany
Hadi Salmasian
Department of Mathematics and Statistics
University of Ottawa
585 King Edward Avenue
Ottawa, ON K1N 6N5
Canada