On Matricial Order Operator Spaces

TL;DR

This paper explores the theory of matricial order operator spaces, generalizing operator systems by incorporating both matricial norms and order, and develops duality theory for these structures.

Contribution

It introduces duality theory for matricial order operator spaces and defines properties like normality and generation, advancing the understanding of their structure.

Findings

Duality between minimal and maximal matricial order structures.

Operator systems and Schatten spaces as examples.

Banach lattices can be equipped with matricial order structures.

Abstract

We investigate the category of ``matricial order operator spaces,'' which generalize operator systems, being equipped with both matricial norms and matricial order. For these objects, we develop duality theory. Taking a cue from the theory of ordered normed spaces, we introduce two important properties describing the interplay between order and norm -- ``normality'' and ``generation,'' and show that they are dual to each other. As examples, we consider operator systems (in particular, C*-algebras), and Schatten spaces. We also describe the minimal and maximal matricial order structures (which, again, turn out to be in duality), and show how Banach lattices can be equipped with such structures.