A Morita characterisation for algebras and spaces of operators on Hilbert spaces

TL;DR

This paper develops a Morita theory framework for operator algebras and spaces, characterising their equivalences through categories of modules and algebraic extensions, with implications for unital operator spaces.

Contribution

It introduces $ riangle$ and $\sigma riangle$-pairs for operator algebras and characterises their Morita equivalences via module categories and algebraic extensions.

Findings

$ riangle$-pairs characterised by module categories

$\sigma riangle$-Morita equivalence implies stable isomorphism

Results on Morita equivalence of unital operator spaces

Abstract

We introduce the notion of $\Delta$ and $\sigma\,\Delta-$ pairs for operator algebras and characterise $\Delta-$ pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of $\Delta$-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that $\sigma\Delta$-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to $\Delta$-Morita equivalence of their algebraic extensions.