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Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

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Abstract

We prove that the relationE K σ is Borel reducible to isomorphism and complemented biembeddability between subspaces ofc 0 orl p with 1≤p<2. We also show that the relationE K σ ⊗=+ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz isomorphism between subspaces ofL p for 1≤p<2.

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References

  1. R. Anisca,Unconditional decompositions in subspaces of l 2(X), Positivity8 (2004), 253–274.

    Article  MathSciNet  Google Scholar 

  2. H. Becker and A. S. Kechris,The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, 232, Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

  3. B. Bossard,A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fundamenta Mathematicae172 (2002), 117–152.

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Burlando,On subspaces of direct sums of infinite sequences of Banach spaces, Atti della Accademia Ligure di Scienze e Lettere46 (1989), 96–105.

    MathSciNet  Google Scholar 

  5. P. G. Casazza and N. J. Kalton,Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces, Israel Journal of Mathematics95 (1996), 349–373.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. G. Casazza and N. J. Kalton,Uniqueness of unconditional bases in Banach spaces, Israel Journal of Mathematics103 (1998), 141–175.

    MATH  MathSciNet  Google Scholar 

  7. P. G. Casazza and N. J. Kalton,Uniqueness of unconditional bases in c 0-Products, Studia Mathematica133 (1999), 275–294.

    MATH  MathSciNet  Google Scholar 

  8. J. Feldman, P. Hahn and C. C. Moore,Orbit structure and countable sections for actions of continuous groups, Advances in Mathematics28 (1978), 186–230.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. Ferenczi and E. M. Galego,Some results about the Schroeder-Bernstein property for separable Banach spaces, Canadian Journal of Mathematics, to appear.

  10. V. Ferenczi and C. Rosendal,On the number of non isomorphic subspaces of a Banach space, Studia Mathematica168 (2005), 203–216.

    MATH  MathSciNet  Google Scholar 

  11. V. Ferenczi and C. Rosendal,Ergodic Banach spaces, Advances in Mathematics, to appear.

  12. H. Friedman and L. Stanley,A Borel reducibility theory for classes of countable structures, Journal of Symbolic Logic54 (1989), 894–914.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. T. Gowers,A solution to the Schroeder-Bernstein problem for Banach space, The Bulletin of the London Mathematical Society28 (1996), 297–304.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. T. Gowers,An infinite Ramsey theorem and some Banach-space dichotomies, Annals of Mathematics (2)156 (2002), 797–833.

    MATH  MathSciNet  Google Scholar 

  15. L. A. Harrington, A. S. Kechris and A. Louveau,A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society3 (1990), 903–928.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Heinrich and P. Mankiewicz,Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Mathematica73 (1982), 225–251.

    MATH  MathSciNet  Google Scholar 

  17. C. W. Henson, J. Iovino, A. S. Kechris and E. Odell,Analysis and Logic, Lectures from the mini-courses offered at the International Conference held at the University of Mons-Hainaut, 1997, London Mathematical Society Lecture Note Series, 262, Cambridge University Press, Cambridge, 2002.

    Google Scholar 

  18. S. Jackson, A. S. Kechris and A. Louveau,Countable Borel equivalence relations, Journal of Mathematical Logic2 (2002), 1–80.

    Article  MATH  MathSciNet  Google Scholar 

  19. N. J. Kalton,Lattice structures on Banach spaces, Memoirs of the American Mathematical Society493 (1993).

  20. N. J. Kalton,A remark on Banach spaces isomorphic to their squares, inFunction Spaces (Edwardsville, IL), Contemporary Mathematics232 (1998), 211–217.

    MathSciNet  Google Scholar 

  21. A. S. Kechris,Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

  22. A. S. Kechris,New directions in descriptive set theory, Bulletin of Symbolic Logic5 (1999), 161–174.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. S. Kechris and A. Louveau,The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society10 (1997), 215–242.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. A. Komorowski and N. Tomczak-Jaegermann,Banach spaces without local unconditional structure, Israel Journal of Mathematics89 (1995), 205–226;Erratum to “Banach spaces without local unconditional structure”, Israel Journal of Mathematics105 (1998), 85–92.

    MATH  MathSciNet  Google Scholar 

  25. J. Lindentrauss and L. Tzafriri,Classical Banach Spaces, Springer-Verlag, New York-Heidelberg-Berlin, 1979.

    Google Scholar 

  26. A. Louveau,On the reducibility order between Borel equivalence relations, inLogic, Methodology and Philosophy of Science, IX (Uppsala, 1991), Studies in Logic and the Foundations of Mathematics134, North-Holland, Amsterdam, 1994, pp. 151–155.

    Chapter  Google Scholar 

  27. A. Louveau and C. Rosendal,Relations d’équivalence analytiques complètes, Comptes Rendus de l’Académie des Sciences, Paris, Série I333 (2001), 903–906.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Maurey,Type, cotype and K-convexity, inHandbook of the Geometry of Banach Spaces (W. B. Johnson and J. Lindenstrauss, eds.), Vol. 2, North-Holland, Amsterdam, 2003, pp. 1299–1332.

    Chapter  Google Scholar 

  29. C. Rosendal,Etude descriptive de l’isomorphisme dans la classe des espaces de Banach, Thèse de doctorat de l’Université Paris 6, 2003.

  30. C. Rosendal,Incomparable, non-isomorphic and minimal Banach spaces, Fundamenta Mathematicae183 (2004), 253–274.

    MATH  MathSciNet  Google Scholar 

  31. C. Rosendal,Cofinal families of Borel equivalence relations and quasiorders, preprint.

  32. J. H. Silver,Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Mathematical Logic18 (1980), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Szankowski,Subspaces without the approximation property, Israel Journal of Mathematics30 (1978), 123–129.

    MATH  MathSciNet  Google Scholar 

  34. N. Tomczak-Jaegermann,Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics 38, Longman Sci. Tech., Harlow, 1989.

    MATH  Google Scholar 

  35. N. Tomczak-Jaegermann,A solution of the homogeneous Banach space problem, Canadian Mathematical Society, 1945–1995,3, Canadian Mathematical Society, Ottawa, ON, 1996, pp. 267–286.

    Google Scholar 

  36. P. Wojtaszczyk,On projections and unconditional bases in direct sums of Banach spaces. II, Studia Mathematica62 (1978), 193–201.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Equipe d’Analyse Fonctionnelle, Université Paris 6, Boîte 186, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Valentin Ferenczi

  2. Department of Mathematics, IME, University of São Paulo, 05311-970, São Paulo, SP, Brasil

    Elói Medina Galego

Authors
  1. Valentin Ferenczi
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  2. Elói Medina Galego
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Correspondence to Valentin Ferenczi.

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This author was supported by FAPESP Grant 2002/09662-1.

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Ferenczi, V., Galego, E.M. Some equivalence relations which are borel reducible to isomorphism between separable banach spaces. Isr. J. Math. 152, 61–82 (2006). https://doi.org/10.1007/BF02771976

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  • DOI: https://doi.org/10.1007/BF02771976

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