From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

Abstract

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 {w_{G}Z_{G}} of its space Z G {Z_{G}} of minimal prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G {X_{G}} of maximal ideals, and the well-known continuous surjection of Z G {Z_{G}} onto X G {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 {Z_{G}} – namely, the sublattice subgroup of C ⁡ ( Z G ) {\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G {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.