Archimedean l-groups with strong unit: cozero-sets and coincidence of types of ideals

TL;DR

This paper investigates the structure of archimedean l-groups with strong units, focusing on the relationships between different types of ideals and their topological properties, using the Yosida representation and cozero-sets.

Contribution

It characterizes when principal ideals and polars coincide as W*-kernels in archimedean l-groups with strong units, linking algebraic and topological properties.

Findings

Equivalence of ideal properties (M), (Y), and (CR) under certain conditions

Characterization of ideals via cozero-sets and regular open sets

Topological perspective sharpens understanding of ideal structures

Abstract

$\bf{W}^*$ is the category of the archimedean l-groups with distinguished strong order unit and unit-preserving l-group homomorphisms. For $G \in \bf{W}^*$, we have the canonical compact space $YG$, and Yosida representation $G \leq C(YG)$, thus, for $g \in G$, the cozero-set coz(g) in $YG$. The ideals at issue in $G$ include the principal ideals and polars, $G(g)$ and $g^{\perp \perp}$, respectively, and the $\bf{W}^*$-kernels of $\bf{W}^*$-morphisms from $G$. The ``coincidences of types" include these properties of $G$: (M) Each $G(g) = g^{\perp \perp}$; (Y) Each $G(g)$ is a $\bf{W}^*$-kernel; (CR) Each $g^{\perp \perp}$ is a $\bf{W}^*$-kernel (iff each coz(g) is regular open). For each of these, we give numerous ``rephrasings", and examples, and note that (M) = (Y) $\cap$ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on $\bf{W}^*$ brings topology to bear, and sharpens the view.