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 T+(E) and 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 infinity(E) and H infinity(F). We give special attention to W-graph correspondences and show a number of results related to their Morita equivalence.