$C^{\ast }$-algebras and direct integral decomposition for Lie supergroups

Abstract

For every finite dimensional Lie supergroup $(G,\mathfrak{g})$, we define a $C^{\ast }$-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 $\ast$-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 $C^{\ast }$-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.

References in corpus (3)

• Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles
• Unitary representations of nilpotent super Lie groups
• Harmonic analysis on Heisenberg--Clifford Lie supergroups