Morita Equivalence of W*-Correspondences and their Hardy Algebras

Abstract

Muhly and Solel developed a notion of Morita equivalence for $C^{\ast }$- correspondences, which they used to show that if two $C^{\ast }$-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${}^{\ast }$ version of this result by considering (weak) Morita equivalence of $W^{\ast }$-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of $W^{\ast }$-correspondences $E$ and $F$ implies weak Morita equivalence of their Hardy algebras $H^{\infty }(E)$ and $H^{\infty }(F)$. We give special attention to $W^{\ast }$-graph correspondences and show a number of results related to their Morita equivalence.

Complex Anal. Oper. Theory (2019)

Cited by in corpus (1)

• On Automorphism Groups of Hardy Algebras