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Representations of Archimedean Riesz Spaces by Continuous Functions

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Positive Operators and Semigroups on Banach Lattices
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Abstract

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces

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References

  1. Y.A. Abramovich, A.I. Veksler, A.V. Koldunov, On operators preserving disjointness, Soviet Math. Dokl. 20 (1979), pp. 1089–1093.

    MATH  Google Scholar 

  2. S.J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), pp. 599–631.

    Article  MathSciNet  Google Scholar 

  3. S.J. Bernau, A note on 4-spaces, Math. Ann. 200 (1973), pp. 281–286.

    Article  MathSciNet  MATH  Google Scholar 

  4. S.J. Bernau, Orthomorphisms of Archimedean vector lattices, Proc. Camb. Phil. Soc. 89 (1979), pp. 119–128.

    Article  MathSciNet  Google Scholar 

  5. H.F. Bohnenblust, On axiomatic characterisation of Lp spaces, Duke Math. J. 6 (1940), pp. 627–640.

    Article  MathSciNet  Google Scholar 

  6. H.F. Bohnenblust, S. Kakutani, Concrete representations of (M)-spaces, Ann. Math. 42 (1941), pp. 1025–1028.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stables et espaces LP, Ann. Inst. H. Poincare 2 (1965/6), pp. 231–259.

    Google Scholar 

  8. E.B. Davies, The structure and ideal theory of the predual of a Banach lattice, Trans. Amer. Math. Soc. 131 (1968), pp. 544–555.

    Article  MathSciNet  MATH  Google Scholar 

  9. E.B. Davies, The Choquet theory and representation of ordered Banach spaces, Illinois J. Math. 13 (1969), pp. 176–187.

    MathSciNet  MATH  Google Scholar 

  10. D.H. Fremlin, Abstract Kothe spaces II, Proc. Cam. Phil. Soc. 63 (1967), pp. 951–956.

    Article  MathSciNet  Google Scholar 

  11. H. Gordon, Measures defined by abstract L p spaces, Pacific J. Math. 10 (1960), pp. 557–562.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Goullet de Rugy, La théorie des cônes biréticulés, Ann. Inst. Fourier (Grenoble) 21 (1971), pp. 1–18.

    Article  MATH  Google Scholar 

  13. A. Goullet de Rugy, La structure ideale des M-espaces, J. Maths. Pures et Appl. 51 (1972), pp. 331–373.

    MATH  Google Scholar 

  14. R. Haydon, Injective Banach lattices, Math. Z. 156 (1977), pp. 19–47.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Kakutani, Concrete representation of abstract L-spaces and the mean ergodic theorem, Ann. Math. 42 (1941), pp. 523–537.

    Article  MathSciNet  Google Scholar 

  16. S. Kakutani, Concrete representations of abstract M-spaces, Ann. Math. 42 (1941), pp. 994–1024.

    Article  MathSciNet  MATH  Google Scholar 

  17. L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, Functional Analysis in Partially Ordered Spaces (Russian), Gostekhizdat, Moscow, 1950.

    Google Scholar 

  18. M. Klein, S. Krein, On an inner characterisation of the set of all continuous functions defined on a bicompact Hausdorff space, C.R. Acad. Sci. URSS 27 (1940), pp. 427–430.

    Google Scholar 

  19. H.E. Lacey, S.J. Bernau, Characterisations and classifications of some classical Banach spaces, Ade. in Math. 12 (1974), pp. 367–401.

    Article  MathSciNet  MATH  Google Scholar 

  20. H.P. Lotz, Zur Idealstruktur in Banachverbänden, Habilitationsschrii t Tubingen (1969).

    Google Scholar 

  21. W. A. J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London, 1971.

    MATH  Google Scholar 

  22. W.A.J. Luxemburg, Some Aspects of the Theory of Riesz Spaces, Univ. of Arkansas Lecture Notes, Fayetteville (1979).

    Google Scholar 

  23. F. Maeda, T. Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. 12 (1942), pp. 17–35.

    MathSciNet  MATH  Google Scholar 

  24. J.T. Marti, Topological representations of abstract Lp spaces, Math. Ann. 185 (1970), pp. 315–321.

    Article  MathSciNet  MATH  Google Scholar 

  25. P.T.N. McPolin, Disjointness preserving linear mappings on a vector lattice, Ph.D. Thesis, Q.U.B. (1983).

    Google Scholar 

  26. P.T.N. McPolin, A.W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Pmc. Cam. Phil. Soc. 97 (1985), pp. 481–487.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Meyer, Representations des espaces vectoriels réticulés, Seminaire Chuquet 13 (1973/4), pp. 1–12.

    Google Scholar 

  28. M. Meyer, Quelques propriétés des homomorphismes d’espaces vectoriels réticulés, E.R.A. Université Paris VI 294 (1978).

    Google Scholar 

  29. R.J. Nagel, Darstellung von Verbandsoperatoren auf Banach- verbänden, Rev. Acad. Ci. Zaragoza 27 (1972), pp. 281–288.

    MathSciNet  MATH  Google Scholar 

  30. R.J. Nagel, Ordnungstetigkeit in Banachverbänden, Manuscripta Math 9 (1973), pp. 9–27.

    Article  MathSciNet  MATH  Google Scholar 

  31. R.J. Nagel, A Stone-Weierstrass theorem for Banach lattices, Studia Math. 47 (1973), pp. 75–82.

    MathSciNet  MATH  Google Scholar 

  32. H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes, Proc. Imp. Acad. Tokyo 17 (1941), pp. 301–307.

    Article  MathSciNet  Google Scholar 

  33. H. Nakano, Über die Charakterisierung des allgemeinen C-Raumes II, Proc. Imp. Acad. Tokyo 18 (1942), pp. 280–286.

    Article  MathSciNet  Google Scholar 

  34. H. Nakano, Stetige lineare Funktionale auf dem teilweisgeordneten Modul, J. Fac. Sci. Imp. Univ. Tokyo 4 (1942), pp. 201–382.

    MATH  Google Scholar 

  35. H. Nakano, Über das System aller stetiger Funktionen auf ein topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), pp. 308–310.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Ogasawara, Theory of vector lattices I, J. Sci. Hiroshima Univ. 12 (1942), pp. 37–100.

    MathSciNet  Google Scholar 

  37. T. Ogasawara, Theory of vector lattices II, J. Sci. Hiroshima Univ. 13 (1944), pp. 41–161.

    MathSciNet  Google Scholar 

  38. H.H. Schaefer, On the representation of Bauach lattices by continuous numerical functions, Math. Z. 125 (1972), pp. 215–232.

    Article  MathSciNet  MATH  Google Scholar 

  39. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag. BerlinHeidelberg-New York, 1974.

    Book  MATH  Google Scholar 

  40. B.Z. Vulikh, Concrete representation of partially ordered linear spaces (Russian), Doklady Akad. Nauk USSR 58 (1947), pp. 733–736.

    Google Scholar 

  41. B.Z. Vulikh, On concrete representation of partially ordered lineals (Russian), Doklady Akad. Nauk USSR 78 (1951), pp. 189–192.

    Google Scholar 

  42. B.Z. Vulikh, Some topics in the theory of partially ordered linear spaces (Russian), Izvestia AN USSR see. math. 17 (1953), pp. 365–388.

    Google Scholar 

  43. B.Z. Vulikh, G.Y. Lozanovskii, On representation of order continuous and regular functionals on partially ordered spaces (Russian), Mat. Sbornik 84 (1971), pp. 331–352.

    Google Scholar 

  44. A.W. Wickstead, Representation and duality of multiplication operators on Archimedean Riesz spaces, Compositio Math. 35 (1977), pp. 225–238.

    MathSciNet  MATH  Google Scholar 

  45. K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 18 (1941/2), pp. 339–342.

    Google Scholar 

  46. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.

    MATH  Google Scholar 

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Wickstead, A.W. (1992). Representations of Archimedean Riesz Spaces by Continuous Functions. In: Huijsmans, C.B., Luxemburg, W.A.J. (eds) Positive Operators and Semigroups on Banach Lattices. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2721-1_13

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  • DOI: https://doi.org/10.1007/978-94-017-2721-1_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4205-7

  • Online ISBN: 978-94-017-2721-1

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