Hilbert $C^{\ast }$-modules over ${\Sigma }^{\ast }$-algebras II: ${\Sigma }^{\ast }$-Morita equivalence

Abstract

In previous work, we defined and studied ${\Sigma }^{\ast }$-modules, a class of Hilbert $C^{\ast }$-modules over ${\Sigma }^{\ast }$-algebras (the latter are $C^{\ast }$-algebras that are sequentially closed in the weak operator topology). The present work continues this study by developing the appropriate ${\Sigma }^{\ast }$-algebraic analogue of the notion of strong Morita equivalence for $C^{\ast }$-algebras. We define strong ${\Sigma }^{\ast }$-Morita equivalence, prove a few characterizations, look at the relationship with equivalence of categories of a certain type of Hilbert space representation, study ${\Sigma }^{\ast }$-versions of the interior and exterior tensor products, and prove a ${\Sigma }^{\ast }$-version of the Brown-Green-Rieffel stable isomorphism theorem.