On Automorphism Groups of Hardy Algebras

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.