Schur Class Operator Functions and Automorphisms of Hardy Algebras
Abstract
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 Picklike 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606672
 Bibcode:
 2006math......6672M
 Keywords:

 Mathematics  Operator Algebras;
 47A57;
 47L55;
 47L65;
 47L75;
 46L08;
 46L53;
 47A48;
 46L52;
 47L30;
 46T25