On Automorphism Groups of Hardy Algebras

TL;DR

This paper investigates the automorphism groups of Hardy algebras associated with W*-correspondences, providing matrix representations and exploring their algebraic structures and applications to Morita equivalence.

Contribution

It introduces a matrix representation for automorphism groups of Hardy algebras and the unit disc of intertwiners, advancing understanding of their algebraic properties and applications.

Findings

Matrix representation of automorphism groups

Descriptions of algebraic structure features

Application to Morita equivalence of W*-correspondences

Abstract

Let $E$ be a $W^{*}$-correspondence and let $H^{\infty}(E)$ be the associated Hardy algebra. The unit disc of intertwiners $\mathbb{D}((E^{\sigma})^{*})$ plays a central role in the study of $H^{\infty}(E)$. We show a number of results related to the automorphism groups of both $H^{\infty}(E)$ and $\mathbb{D}((E^{\sigma})^{*})$. We find a matrix representation for these groups and describe several features of their algebraic structure. Furthermore, we show an application of $Aut(\mathbb{D}({(E^{\sigma}})^*))$ to the study of Morita equivalence of $W^{*}$-correspondences.