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
... [Show full abstract] 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.