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