# Schur class operator functions and automorphisms of Hardy algebras

### Paul S. Muhly

### Baruch Solel

## Abstract

Let $E$ be a $W_{∗}$-correspondence over a von Neumann algebra $M$ and let $H_{∞}(E)$ be the associated Hardy algebra. If $σ$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E_{σ}$ and represent elements in $H_{∞}(E)$ as $B(H)$-valued functions on the unit ball $D(E_{σ})_{∗}$. 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 $H_{∞}(E)$ in terms of special Möbius transformations on $D(E_{σ})$. Particular attention is devoted to the $H_{∞}$-algebras that are associated to graphs.

## Cite this article

Paul S. Muhly, Baruch Solel, Schur class operator functions and automorphisms of Hardy algebras. Doc. Math. 13 (2008), pp. 365–411

DOI 10.4171/DM/250