Morita equivalence of dual operator algebras

TL;DR

This paper develops a new framework called weak Morita equivalence for dual operator algebras, extending Morita theory to the setting of weak* closed operator algebras on Hilbert spaces.

Contribution

It introduces variants of Morita equivalence tailored for dual operator algebras, expanding the theory to nonselfadjoint and weak* closed algebra contexts.

Findings

Established new variants of strong Morita equivalence for dual algebras

Extended Rieffel's $W^*$-algebraic Morita equivalence to nonselfadjoint cases

Provided foundational results for weak Morita equivalence in operator algebra theory

Abstract

We consider a variant of the notion of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators, which we call {\em weak Morita equivalence}. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint variants of aspects of Rieffel's $W^*$-algebraic Morita equivalence.