Stone-Gelfand duality for metrically complete lattice-ordered groups

TL;DR

This paper generalizes Stone-Gelfand duality to metrically complete lattice-ordered groups, introducing a new class of 'arithmetic' compact spaces and extending classical duality results beyond real vector spaces.

Contribution

It develops a duality for metrically complete unital lattice-ordered groups without requiring them to be real vector spaces, using a novel notion of arithmetic compact Hausdorff spaces.

Findings

Extended duality theorem for these groups.

Generalized Urysohn's Lemma for arithmetic spaces.

Connections with the theory of dimension groups.

Abstract

We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing $1$, namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the whole of $\mathbb{R}$. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic" compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces whose each point is assigned the entire group of real numbers. In the introduction we indicate motivations from and connections with the theory of dimension groups.