  To each irreducible infinite dimensional representation $(\pi,\cH)$ of a
$C^*$-algebra $\cA$, we associate a collection of irreducible norm-continuous
unitary representations $\pi_{\lambda}^\cA$ of its unitary group $\U(\cA)$,
whose equivalence classes are parameterized by highest weights in the same way
as the irreducible bounded unitary representations of the group $\U_\infty(\cH)
= \U(\cH) \cap (\1 + K(\cH))$ are. These are precisely the representations
arising in the decomposition of the tensor products $\cH^{\otimes n} \otimes
(\cH^*)^{\otimes m}$ under $\U(\cA)$. We show that these representations can be
realized by sections of holomorphic line bundles over homogeneous K\"ahler
manifolds on which $\U(\cA)$ acts transitively and that the corresponding
norm-closed momentum sets $I_{\pi_\lambda^\cA}^{\bf n} \subeq \fu(\cA)'$
distinguish inequivalent representations of this type.
