Schur Class Operator Functions and Automorphisms of Hardy Algebras

Details Schur Class Operator Functions and Automorphisms of Hardy Algebras Schur Class Operator Functions and Automorphisms of Hardy Algebras

2006-06-27

arxiv.org/abs/math/0606672v2

Mathematics

Operator Algebras

Minor corrections. A few comments and references added

Scientific paper

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 M\"{o}bius transformations on $\mathbb{D}(E^{\sigma})$. Particular attention is devoted to the $H^{\infty}% $-algebras that are associated to graphs.

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