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- Title
From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra.
- Authors
Ball, Richard N.; Marra, Vincenzo; McNeill, Daniel; Pedrini, Andrea
- 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 <italic>G</italic> with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate <italic>G</italic>-indexed zero-dimensional compactification w G Z G {w_{G}Z_{G}} of its space Z G {Z_{G}} of <italic>minimal</italic> prime ideals.The two further ingredients needed to establish this representation are the Yosida representation of <italic>G</italic> on its space X G {X_{G}} of <italic>maximal</italic> 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 <italic>G</italic> 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 <italic>G</italic> along with all characteristic functions of clopen (closed and open) subsets of Z G {Z_{G}} which are determined by elements of <italic>G</italic> – is precisely the classical projectable hull of <italic>G</italic>.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.
- Subjects
RIESZ spaces; LATTICE theory; MAXIMAL ideals; CONTINUOUS functions; FREUDENTHAL compactification; SUBGROUP growth
- Publication
Forum Mathematicum, 2018, Vol 30, Issue 2, p513
- ISSN
0933-7741
- Publication type
Article
- DOI
10.1515/forum-2017-0044