The Archimedean order unitization of seminormed ordered vector spaces

TL;DR

This work constructs a minimal Archimedean order unitization $E_1$ of any seminormed preordered vector space $(E,E_+,p)$ that simultaneously Archimedeanizes and adjoins an order unit via $E_1 = (E/N) \oplus \mathbb{R}$ with $(E_1)_+ = \{(x+N,\lambda): d(x,E_+)\le \lambda\}$ and $u=(0+N,1)$, where $N=\overline{E_+}\cap -\overline{E_+}$ and $d(x,E_+)=\inf_{y\in E_+} p(x-y)$. The authors establish a universal property: for every AOU space $F$ and contractive positive map $\psi:E\to F$, there exists a unique unital positive extension $\psi_1:E_1\to F$ with $\psi=\psi_1\circ \phi$, and provide multiple characterizations of $E_1$ (duality via $E'_+$ with $\|\varphi\|\le 1$, and a representation in terms of a compact $\Omega$). They develop the 1-max-normalization $p_u(x)=\lVert \phi(x)\rVert_u=\max(d(x,E_+),d(-x,E_+))$, show it is the largest $1$-max-normal seminorm below $p$, and relate it to local fullness and norm-equivalence. The paper then specializes to two key examples: the unitization of an order unit space, yielding $E_1\cong E_{Arch,1}\oplus\mathbb{R}$, and the self-adjoint part of a C*-algebra, yielding $E_1\cong \tilde{\mathcal{A}}^{sa}$. It further connects these ideas to representation theory via Kadison representations and a direct $E\to C(\Omega_{w^*})$ construction, and applies the framework to matrix ordered operator spaces, giving a geometrically transparent simplification of Werner's partial unitization and a sharp criterion for when $E\to E^\sharp$ is a complete isomorphism in terms of $\kappa$-max-normality of the matrix norms. Overall, the work unifies Archimedeanization and unitization across classical and operator-algebraic settings, provides practical normalization tools for seminorms, and yields concrete criteria for non-unital operator-system structures.

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.

Figures (3)

• Figure 1: The base of the positive cone of the minimal Archimedean order unitization of $(\mathbb{R}^2,\mathbb{R}_{\geq 0}^2,\lVert \:\cdot\: \rVert_{\ell_q})$ in the $z = 1$ plane, for different values of $q$.
• Figure 2: The state space of the Archimedean order unitization of $(\mathbb{R}^2,\mathbb{R}_{\geq 0}^2,\lVert \:\cdot\: \rVert_{\ell_q})$, drawn in the $z = 1$ plane, for different values of $q$. (Compare \ref{['fig:R2-example']}.)
• Figure 3: The unit ball of the $1$-max-normalization of $(\mathbb{R}^2,\mathbb{R}_{\geq 0}^2,\lVert \:\cdot\: \rVert_{\ell_q})$, for different values of $q$.

Theorems & Definitions (88)

• theorem 1: Paulsen and Tomforde, Paulsen-Tomforde
• theorem 2: Emelyanov, Emelyanov
• theorem 3
• theorem 4
• theorem 5
• corollary 6
• proposition 7
• proposition 8: compare Kadison-representation
• proof
• definition 3.1