Schur--Weyl Theory for $C^*$-algebras

Abstract

To each irreducible infinite dimensional representation $(π,\cH)$ of a $C^*$-algebra $\cA$, we associate a collection of irreducible norm-continuous unitary representations $π_λ^\cA$ 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 Kähler manifolds on which $\U(\cA)$ acts transitively and that the corresponding norm-closed momentum sets $I_{π_λ^\cA}^{\bf n} \subeq \fu(\cA)'$ distinguish inequivalent representations of this type.