Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\pi ,\mathcal {H})$\end{document} of a C*‐algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}$\end{document}, we associate a collection of irreducible norm‐continuous unitary representations \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\pi _{\lambda }^\mathcal {A}$\end{document} of its unitary group \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\rm U}(\mathcal {A})$\end{document}, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$\end{document} are. These are precisely the representations arising in the decomposition of the tensor products \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$\end{document} under \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\rm U}(\mathcal {A})$\end{document}. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\rm U}(\mathcal {A})$\end{document} acts transitively and that the corresponding norm‐closed momentum sets \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$\end{document} distinguish inequivalent representations of this type.