Morita equivalence of w*-rigged modules

TL;DR

This paper introduces new Morita equivalence concepts for $w^{ riangledown}$-rigged modules over dual operator algebras, exploring their implications for module isomorphism and applying the theory to nest algebras.

Contribution

It proposes two novel Morita equivalence notions for $w^{ riangledown}$-rigged modules and analyzes their relationship with stable isomorphism, especially over nest algebras.

Findings

New Morita equivalence notions are introduced.

These notions are examined for their implications on stable isomorphism.

Application to nest algebras characterizes the equivalence in that context.

Abstract

The $w^{\star}$-rigged modules over dual operator algebras were introduced by Blecher and Kashyap as a generalization of $W^{\star}$-modules. In this paper, we introduce two new types of Morita equivalence between right $w^{\star}$-rigged modules over unital dual operator algebras and we examine whether these notions imply stable isomorphism between the corresponding modules. Furthermore we investigate them in detail for the class of right $w^{\star}$-rigged modules over nest algebras, that was characterised by G.K.Eleftherakis.