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. We study how different representations of a W*-algebra give rise to Morita equivalent objects. For example, we show that if (E,A) is a W*-graph correspondence and we have two faithful normal representations $\sigma$ and $\tau$ of A, then the commutants of the induced representions $\sigma ^{\ms{F}(E)}(H^{\infty}(E))$ and $\tau ^{\ms{F}(E)}(H^{\infty}(E))$ are weakly Morita equivalent dual operator algebras.
We also develop a categorical approach to Morita equivalence of W*- correspondences. This involves building categories of covariant representations and studying the groups $Aut(\mathbb{D}({(E^{\sigma}})^*)$ and $Aut(H^{\infty}(E))$ (the automorphism groups of the unit ball of intertwiners and the Hardy algebra). In this regard, we advance the work of Muhly and Solel by showing new results about these groups, their matrix representation and their algebraic properties.