Schur class operator functions and automorphisms of Hardy algebras

Muhly, Paul S. ; Solel, Baruch

Documenta mathematica, Tome 13 (2008), pp. 365-411.

Résumé

Summary: Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ be the associated Hardy algebra. If $\sigma$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E^{\sigma}$ and represent elements in $H^{\infty}(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^{\sigma})^{\ast}$. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of $H^{\infty}(E)$ in terms of special Möbius transformations on $\mathbb{D}(E^{\sigma})$. Particular attention is devoted to the $H^{\infty}$-algebras that are associated to graphs.

Détail

Classification : 46E22, 46E50, 46G20, 46H15, 46H25, 46K50, 46L08, 46L89
Keywords: Hardy algebras, tensor algebras, Schur class functions, $W^*$-correspondence, noncommutative realization theory, Möbius transformations, free semigroup algebras, graph algebras, Nevanlinna-Pick interpolation