Nonunital Operator Systems as Modules in Enriched Category Theory

An operator system is similar to a module over a ring, with the role of scalar multiplication played by the action of completely positive maps. Using enriched category theory, we make this analogy into a precise categorical equivalence, namely between a certain category of nonunital operator systems and a certain category of left modules over the category of matrix algebras enriched over regularly ordered Banach spaces. Using right modules instead yields an equivalence with a certain category of nonunital dual operator systems.
We also develop general separation, representation and extension theorems for modules in enriched category theory. Specializing these to our nonunital operator systems recovers results which partly recover the corresponding classical theorems for operator systems.