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