# Hilbert $C^*$-modules over $\Sigma^*$-algebras II: $\Sigma^*$-Morita equivalence

## TL;DR

This paper extends the theory of $oldsymbol{ ext{Hilbert } C^* ext{-modules}}$ over $oldsymbol{ ext{$oldsymbol{ ext{ extSigma}^*}$-algebras}}$ by developing a $oldsymbol{ ext{Morita}}$-type equivalence framework, including tensor products and stability results.

## Contribution

It introduces the concept of strong $ ext{ extSigma}^*$-Morita equivalence and explores its properties, relationships, and applications in the context of $ ext{ extSigma}^*$-algebras.

## Key findings

- Defined strong $ ext{ extSigma}^*$-Morita equivalence.
- Characterized the equivalence and related it to category of Hilbert space representations.
- Proved a $ ext{ extSigma}^*$-version of the Brown-Green-Rieffel stable isomorphism theorem.

## Abstract

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