  We study a generalization of the results \in \cite{cfk} to the case of
$SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its
finite dimensional complex irreducible representations, we give a reducibility
result for representations not containing the trivial character, and we compute
explicitly the corresponding matrix elements. In the end we give the Peter-Weyl
theorem for $S^{1|2}$.
