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

TL;DR

This paper extends Schur--Weyl theory to $C^*$-algebras by classifying irreducible representations of their unitary groups via highest weights, linking them to tensor product decompositions and geometric realizations.

Contribution

It introduces a new classification of irreducible unitary representations of $C^*$-algebras' unitary groups using highest weights, connecting algebraic and geometric perspectives.

Findings

Representations are parameterized by highest weights similarly to classical cases.

Tensor product decompositions correspond to these representations.

Momentum sets distinguish inequivalent representations.

Abstract

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.