Matrix-ordered Duals of Operator Systems and Projective Limits

TL;DR

This paper explores the duality between inductive and projective limits in categories of operator systems and order unit spaces, extending finite-dimensional duality results to the separable case.

Contribution

It constructs projective limits in these categories and proves their duality with inductive limits under certain conditions, generalizing previous finite-dimensional results.

Findings

Established duality between inductive and projective limits in operator system categories.

Extended matrix-ordered duals from finite-dimensional to separable operator systems.

Provided conditions under which duality holds for these limits.

Abstract

We construct projective limit of projective sequence in the following categories: Archimedean order unit spaces with unital positive maps and operator systems with unital completely positive maps. We prove that inductive limit and projective limit in these categories are in duality, provided that the dual objects remain in the same categories and the maps are order embeddings. We generalize a result of Choi and Effros on matrix-ordered duals of finite-dimensional operator systems to separable case.