A Remark on the Representation of Vector Lattices as Spaces of Continuous Real-Valued Functions

Abstract

The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space Ω, unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice \(\hat L\) of the universally complete vector lattice C ∞(Ω) of all extended-real-valued continuous functions f on Ω for which {ω ∈ Ω: | f(ω)| = ∞} is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that \(\hat L\) consists of bounded functions only.