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

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 $w_G{Z}_G$ 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 $\mathrm{C}(Z_G)$ 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.

19 pages. Major revision. Our previous version contains a mistake: the clopens on the minimal spec of G given by zerosets of principal polars are assumed to be all clopens of the space. We are deeply grateful to an anonymous referee for a counterexample to our previous main statement. Please see the paper for details. Also, minor changes to some proofs, and added examples and references