From Freudenthal's Spectral Theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

TL;DR

This paper extends Freudenthal's Spectral Theorem to represent Archimedean lattice-ordered groups with strong units as function groups on compactifications of their minimal spectra, linking projectable hulls to minimal spectra.

Contribution

It provides the first explicit construction of projectable hulls of unital Archimedean lattice-groups using minimal spectra and compactifications, connecting spectral theory with lattice-ordered group extensions.

Findings

Representation of G via continuous functions on compactification of minimal spectrum

Identification of the projectable hull as a sublattice of continuous functions

Explicit construction linking minimal spectra to projectable hulls

Abstract

We use a landmark result in the theory of Riesz spaces - Freudenthal's 1936 Spectral Theorem - to canonically represent any Archimedean lattice-ordered group $G$ with a strong unit as a (non-separating) lattice-group of real valued continuous functions on an appropriate $G$-indexed zero-dimensional compactification $w_GZ_G$ of its space $Z_G$ of \emph{minimal} prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of $G$ on its space $X_G$ of \emph{maximal} ideals, and the well-known continuous surjection of $Z_G$ onto $X_G$. We then establish our main result by showing that the inclusion-minimal extension of this representation of $G$ that separates the points of $Z_G$ - namely, the sublattice subgroup of ${\rm C}\,(Z_G)$ generated by the image of $G$ along with all characteristic functions of clopen (closed and open) subsets of $Z_G$ which are determined by elements of $G$ - is precisely the classical projectable hull of $G$. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.