  In this paper we determine the projective unitary representations of finite
dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A
\otimes \frak{k}$, where $\frak{k}$ is a compact simple Lie superalgebra and
$A$ is a supercommutative associative (super)algebra; the crucial case is when
$A = \Lambda_s(\mathbb{R})$ is a Gra\ss{}mann algebra. Since we are interested
in projective representations, the first step consists of determining the
cocycles defining the corresponding central extensions. Our second main result
asserts that, if $\frak{k}$ is a simple compact Lie superalgebra with
$\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of
$\Lambda_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary
representation of $\frak{k}$ itself, and these are known by Jakobsen's
classification. If $\frak{k}_1 = \{0\}$, then we likewise reduce the
classification problem to semidirect products of compact Lie groups $K$ with a
Clifford--Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo
and Varadarajan.
