Rene Ardila | Searcharxiv

arXiv2020

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 $\mathbb{D}((E^σ)^{*})$ 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 $\mathbb{D}((E^σ)^{*})$. We find a matrix representation for these groups and describe several features of their algebraic structure. Furthermore, we show an application of $Aut(\mathbb{D}({(E^σ})^*))$ to the study of Morita equivalence of $W^{*}$-correspondences.

math.OA↗

arXiv2019

Morita Equivalence of W*-Correspondences and their Hardy Algebras

Muhly and Solel developed a notion of Morita equivalence for $C^{*}$- correspondences, which they used to show that if two $C^{*}$-correspondences $E$ and $F$ are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak$^{*}$ version of this result by considering (weak) Morita equivalence of $W^{*}$-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of $W^{*}$-correspondences $E$ and $F$ implies weak Morita equivalence of their Hardy algebras $H^{\infty}(E)$ and $H^{\infty}(F)$. We give special attention to $W^{*}$-graph correspondences and show a number of results related to their Morita equivalence.

math.OA↗