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

In: Positive Operators and Semigroups on Banach Lattices

Author

Listed:
  • Yuri A. Abramovich

    (IUPUI, Department of Mathematics)

  • Wolfgang Filter

    (ETH-Zentrum, Mathematik)

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. The aim of this short article is to present a simple complete characterization of such vector lattices.

Suggested Citation

  • Yuri A. Abramovich & Wolfgang Filter, 1992. "A Remark on the Representation of Vector Lattices as Spaces of Continuous Real-Valued Functions," Springer Books, in: C. B. Huijsmans & W. A. J. Luxemburg (ed.), Positive Operators and Semigroups on Banach Lattices, pages 23-26, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-2721-1_2
    DOI: 10.1007/978-94-017-2721-1_2
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