Paul S. Muhly and Baruch Solel

摘要

Let $E$ be a $W^{st}$-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 $mathbb{D}(E^{sigma})^{st}$. 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.