# Morita Equivalence of W*-Correspondences and their Hardy Algebras

## TL;DR

This paper extends Morita equivalence concepts from $C^{*}$-correspondences to $W^{*}$-correspondences, demonstrating that weak Morita equivalence of the correspondences implies weak Morita equivalence of their Hardy algebras, with special focus on $W^{*}$-graph correspondences.

## Contribution

It introduces the weak$^{*}$ version of Morita equivalence for $W^{*}$-correspondences and proves that this equivalence implies the same for their Hardy algebras, expanding the theory to dual operator algebras.

## Key findings

- Weak Morita equivalence of $W^{*}$-correspondences implies weak Morita equivalence of their Hardy algebras.
- Results specialized to $W^{*}$-graph correspondences.
- Established connections between $W^{*}$-correspondences and their Hardy algebras.

## Abstract

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.