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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Functional representation of vector lattices


Author: Isidore Fleischer
Journal: Proc. Amer. Math. Soc. 108 (1990), 471-478
MSC: Primary 06F20; Secondary 46A40, 54C30, 54H12
MathSciNet review: 993750
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Abstract: Every vector lattice is represented in the lattice of distribution functions valued in the complete Boolean algebra of its annihilators; the representation is complete join and positive multiple preserving and subadditive; restricted to the solid vector sublattice without infinitesimals, it preserves the full structure (including any existing infinite lattice extrema) and is faithful. Identifying the distribution with the continuous real-valued functions on the extremally disconnected Stone space of the algebra yields a representation which, specialized to Archimedean vector lattices, embeds them in the densely finite-valued continuous functions; identifying with the equivalence classes of functions measurable for a $ \sigma $-field modulo a $ \sigma $-ideal of "null sets" yields a representation which, specialized to Archimedean vector lattices, embeds them in the classes of a.e. finite functions. This is used to give simple proofs of Freudenthal's spectral theorem and Kakutani's structure theorem for $ L$-spaces.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0993750-3
PII: S 0002-9939(1990)0993750-3
Article copyright: © Copyright 1990 American Mathematical Society