Hilbert $C^*$-modules over $Σ^*$-algebras II: $Σ^*$-Morita equivalence

Abstract

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