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$C^*$-algebras and direct integral decomposition for Lie supergroups (1506.01558v1)
Published 4 Jun 2015 in math.RT
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.