Schur--Weyl Theory for $C^{\ast }$-algebras

To each irreducible infinite dimensional representation $(\pi ,\text{cH})$ of a $C^{\ast }$-algebra $\text{cA}$, we associate a collection of irreducible norm-continuous unitary representations ${\pi }_{\lambda }^{\text{cA}}$ of its unitary group $\text{U}(\text{cA})$, 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 $\text{U}(\text{cA})$ acts transitively and that the corresponding norm-closed momentum sets $I_{{\pi }_{\lambda }^{\text{cA}}}^{\mathbf{n}}\text{subeq}\text{fu}(\text{cA}{)}^{\mathrm{\prime }}$ distinguish inequivalent representations of this type.