$C^*$-algebras and direct integral decomposition for Lie supergroups

TL;DR

This paper constructs a $C^*$-algebra for finite dimensional Lie supergroups, establishing a bijective correspondence with their unitary representations and proving CCR properties for a broad class, including nilpotent and simple supergroups.

Contribution

It introduces a $C^*$-algebra framework for Lie supergroups and demonstrates a correspondence with unitary representations, also proving CCR properties for many classes.

Findings

Established a bijective correspondence between unitary representations and $*$-representations.

Proved the associated $C^*$-algebra is CCR for broad classes of Lie supergroups.

Ensured the uniqueness of direct integral decomposition for these representations.

Abstract

For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathcal A:=\mathcal A(G,\mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,\mathfrak g)$ and nondegenerate $*$-representations of $\mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. 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.