Crossed product algebras and direct integral decomposition for Lie supergroups

Karl-Hermann Neeb and Hadi Salmasian

Vol. 282 (2016), No. 1, 213–232

DOI: 10.2140/pjm.2016.282.213

Abstract

For every finite dimensional Lie supergroup $\left(G,\mathfrak{g}\right)$, we define a $C^{\ast }$-algebra $\mathcal{A}:=\mathcal{A}\left(G,\mathfrak{g}\right)$ and show that there exists a canonical bijective correspondence between unitary representations of $\left(G,\mathfrak{g}\right)$ and nondegenerate $\ast$-representations of $\mathcal{A}$. 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 $C^{\ast }$-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