LAUSR

Muhly and Solel developed a notion of Morita equivalence for $$C^{*}$$C∗-correspondences, which they used to show that if two $$C^{*}$$C∗-correspondences E and F are Morita equivalent then their tensor algebras $${\mathcal {T}}_{+}(E)$$T+(E) and $${\mathcal {T}}_{+}(F)$$T+(F) are (strongly) Morita equivalent operator algebras. We give the weak$$^{*}$$∗ version of this result by considering (weak) Morita equivalence of $$W^{*}$$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 $$W^{*}$$W∗-correspondences E and F implies weak Morita equivalence of their Hardy algebras $$H^{\infty }(E)$$H∞(E) and $$H^{\infty }(F)$$H∞(F). We give special attention to $$W^{*}$$W∗-graph correspondences and show a number of results related to their Morita equivalence.