, Volume 22, Issue 1, pp 301–339 | Cite as

New Axiomatizable classes of Banach spaces via disjointness-preserving isometries

  • Yves Raynaud


Let \(\mathcal {C}\) be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of \(\mathcal {C}\) in their ultrapowers preserve disjointness, the class \(\mathcal {C}^\mathcal {B}\) of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if \(\mathcal {C}\) coincides with its “script class” \(\mathcal {SC}\), so does \(\mathcal {C}^\mathcal {B}\) with \(\mathcal {SC}^\mathcal {B}\). This allows us to give new examples of axiomatizable classes of Banach spaces, namely certain Musielak–Orlicz spaces, Nakano spaces, and mixed norm spaces.


Banach space Banach lattice Axiomatizable class Ultraproducts and ultraroots Disjointness preserving isometries 

Mathematics Subject Classification

Primary 46B42 46E30 Secondary 03C65 46M07 



The author thanks C. W. Henson for his critical reading of a previous version of the manuscript and his suggestions for improving it.


  1. 1.
    Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence (2002)MATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)CrossRefMATHGoogle Scholar
  3. 3.
    Ben Yaacov, I., Berenstein, A., Henson, C. W., Usvyatsov, A.: Model Theory for Metric Structures. Model Theory with Applications to Algebra and Analysis. London Math. Soc. Lecture Note Ser., vol. 2, 350, pp. 315–427. Cambridge Univ. Press (2008)Google Scholar
  4. 4.
    Boulabiar, K., Buskes, G.: Polar decomposition of disjointness preserving operators. Proc. Am. Math. Soc. 132(3), 799–806 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Claas, W.J., Zaanen, A.C.: Orlicz lattices. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. Comment. Math. Special issue 1, 77–93 (1978)Google Scholar
  6. 6.
    Fremlin, D.H.: Measure Theory: Measure Algebras, vol. 3. Torres Fremlin, Colchester (2002)MATHGoogle Scholar
  7. 7.
    Grobler, J.J., Huijsmans, C.B.: Disjointness preserving operators on complex Riesz spaces. Positivity 1, 155–164 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guerre, S., Raynaud, Y.: Sur les isométries de \(L_p(X)\) et le théorème ergodique vectoriel. Can. J. Math. 140, 360–391 (1988)CrossRefMATHGoogle Scholar
  9. 9.
    Heinrich, S.: Ultraproducts in Banach spaces theory. J. Reine Angew. Math. 313, 72–104 (1980)MathSciNetMATHGoogle Scholar
  10. 10.
    Henson, C.W., Moore Jr. L.C.: Nonstandard Analysis and the Theory of Banach Spaces. Nonstandard Analysis–Recent Developments (Victoria, B.C., 1980). Lecture Notes in Math., vol. 983, pp. 27–112. Springer, Berlin (1983)Google Scholar
  11. 11.
    Henson, C.W., Iovino, J.: Ultraproducts in Analysis. Analysis and Logic (Mons 1997). London Math. Soc. Lecture Note Ser., vol. 262, pp. 1–113. Cambridge Univ. Press (2003)Google Scholar
  12. 12.
    Henson, C.W., Raynaud, Y.: On the theory of \(L_p(L_q)\) Banach lattices. Positivity 11, 201–230 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Henson, C.W., Raynaud, Y.: Quantifier elimination in the theory of \(L_p(L_q)\)-Banach lattices. J. Log. Anal. 3, paper 11 (2011)Google Scholar
  14. 14.
    Jamison, J.E., Kamińska, A., Lin, Pei-Kee: Isometries of Musielak–Orlicz spaces II. Studia Math. 104, 75–89 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Johnson, W.B., Lindenstrauss, J., Schechtman, G.: Banach spaces determined by their uniform structures. Geom. Funct. Anal. 6, 430–470 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lacey, H.E.: Local unconditional structure in Banach spaces. Banach spaces of analytic functions. Lecture Notes in Math., vol. 604, pp. 44–56 (1977)Google Scholar
  17. 17.
    Lamperti, J.: On the isometries of certain function-spaces. Pacific J. Math. 8, 459–466 (1958)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Levy, M., Raynaud, Y.: Ultrapuissances des espaces \(L^p(L^q)\). C. R. Acad. Sci. Paris Sér. I Math. 299(3), 81–84 (1984)MathSciNetMATHGoogle Scholar
  19. 19.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II: Function Spaces. Ergebnisse der Math, vol. 97. Springer, Berlin, New-York (1979)Google Scholar
  20. 20.
    Meyer, M.: Le stabilisateur d’un espace vectoriel réticulé. C. R. Acad. Sci. Paris Sér. A 283, 249–250 (1976)MathSciNetMATHGoogle Scholar
  21. 21.
    Meyer, M.: Les homomorphismes d’espaces vectoriels réticulés complexes. C. R. Acad. Sci. Paris Sér. I Math. 292, 793–796 (1981)MathSciNetMATHGoogle Scholar
  22. 22.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  23. 23.
    Poitevin, L.P.: Model theory of nakano spaces. Ph.D. thesis. University of Illinois at Urbana-Champaign (2006)Google Scholar
  24. 24.
    Poitevin, L.P., Raynaud, Y.: Ranges of positive contractive projections in Nakano spaces. Indag. Math. 19, 441–464 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Raynaud, Y.: The range of a contractive projection in \(L_p(H)\). Rev. Mat. Complut. 17, 485–512 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rivera, M.J.: On the classes of \(\cal{L}^\lambda \), quasi-\(\cal{L}^E\) and \(\cal{L}^{\lambda, g}\) spaces. Proc. Am. Math. Soc. 133(7), 2035–2044 (2005)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wnuk, W.: On a representation theorem for convex Orlicz lattices. Bull. Acad. Polon. Sci. Sér. Sci. Math. 28(3–4), 131–136 (1980)MathSciNetMATHGoogle Scholar
  28. 28.
    Wnuk, W.: Representations of Orlicz lattices. Dissertationes Math. (Rozprawy Mat.) 235, 62 (1984)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut Math. Jussieu-Paris-Rive GaucheCNRS, UPMC (Univ. Paris 06) and Univ. Paris-DiderotParis Cedex 05France

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