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Abstract

In previous work, we defined and studied $\varSigma^* $-modules, a class of Hilbert $C^*$-modules over $\varSigma^* $-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 $\varSigma^* $-algebraic analogue of the notion of strong Morita equivalence for $C^*$-algebras. We define strong $\varSigma^*$-Morita equivalence, prove a few characterizations, look at the relationship with equivalence of categories of a certain type of Hilbert space representation, study $\varSigma^*$-versions of the interior and exterior tensor products, and prove a $\varSigma^*$-version of the Brown–Green–Rieffel stable isomorphism theorem.