The Archimedean order unitization of seminormed ordered vector spaces

TL;DR

This paper introduces a method to convert seminormed preordered vector spaces into Archimedean order unit spaces, providing a universal property and applications in ordered vector spaces and matrix ordered operator spaces.

Contribution

It presents a new construction for Archimedean order unitization with a universal property, and offers novel insights into normality criteria and Werner's partial unitization.

Findings

A universal property for the Archimedean order unitization is established.

A simplified internal description of Werner's partial unitization positive cone is provided.

A necessary and sufficient condition for embedding matrix ordered operator spaces as complete isomorphisms is proven.

Abstract

In this paper, we describe a way of turning a seminormed preordered vector space into an Archimedean order unit space. We show that this construction satisfies a universal property similar to that of the Archimedeanization of Paulsen and Tomforde, and we give a number of applications of our result in ordered vector spaces and in matrix ordered operator spaces. In ordered vector spaces, we use our our Archimedean order unitzation to shed new light on normality criteria for seminorms. In matrix ordered operator spaces, we prove several new results about Werner's "partial unitization": we give a simplified "internal" description of the positive cone of Werner's partial unitization, and we prove a necessary and sufficient condition for the embedding of a matrix ordered operator space in its partial unitization to be a complete isomorphism. This last result was already announced in Werner's 2002 paper, but to our knowledge no proof exists in the literature.