2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131107Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ 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 $H^{\infty}(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^σ)^{\ast}$. 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 $\mathbb{D}(E^σ)$. Particular attention is devoted to the $H^{\infty}% $-algebras that are associated to graphs.Minor corrections. A few comments and references addedOperator Algebras47A57, 47L55, 47L65, 47L75, 46L08, 46L53, 47A48, 46L52, 47L30, 46T25text