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 minimal prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of G on its space X_G of 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 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.

From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra / R..N. Ball, V. Marra, D.K. Mc Neill, A. Pedrini. - In: FORUM MATHEMATICUM. - ISSN 0933-7741. - (2017). [Epub ahead of print]

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

V. Marra
;
A. Pedrini
2017

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 minimal prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of G on its space X_G of 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 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.
Freudenthal’s spectral theorem; Riesz space; lattice-ordered group; Archimedean property; Yosida representation; principal projection property; projectable hull; spectral space; maximal ideal; minimal ideal; zero-dimensional space; compactification
Settore MAT/02 - Algebra
Settore MAT/01 - Logica Matematica
2017
ago-2017
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/521168
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