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

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 $sigmaDelta$-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.