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

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