  When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in
the simplicial de Rham complex $\Omega^3(NG)$ which represents the
Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a
central $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whose
Dixmier-Douady class in $\Omega^3(NLSU(2))$ is a kind of transgression of the
second Chern class.
  In this paper, we consider the case of unitary group and construct a central
$U(1)$-extension of $LU(2)$. After that we construct also a cocycle in a
certain triple complex.
