Functional representation of vector lattices. (English) Zbl 0743.46005

Fleischer, Isidore

Proc. Am. Math. Soc. 108, No. 2, 471-478 (1990).

Summary: Every vector lattice is represented in the lattice of distribution functions valued in the complete Boolean algebra of its annihilators; the representation is complete joint 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.

MSC:

46A40	Ordered topological linear spaces, vector lattices
06F20	Ordered abelian groups, Riesz groups, ordered linear spaces
46B40	Ordered normed spaces
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
54C30	Real-valued functions in general topology
54H10	Topological representations of algebraic systems
46E05	Lattices of continuous, differentiable or analytic functions

Keywords:

vector lattice; lattice of distribution functions valued in the complete Boolean algebra of its annihilators; extremally disconnected Stone space; Archimedean vector lattices; densely finite-valued continuous functions; Freudenthal’s spectral theorem; Kakutani’s structure theorem for \(L\)- spaces

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