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The isomorphism class of C0 is not Borel

Israel Journal of Mathematics volume 231, pages 243–268 (2019)Cite this article

Abstract

We show that the class of all Banach spaces which are isomorphic to c0 is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. The proof employs a recent Bourgain–Delbaen construction by Argyros, Gasparis and Motakis.

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References

  1. S. A. Argyros and I. Deliyanni, Examples of asymptotic ℓ1 Banach spaces, Transactions of the American Mathematical Society 349 (1997), 973–995.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. A. Argyros, I. Gasparis and P. Motakis, On the structure of separable L ∞-spaces, Mathematika 62 (2016), 685–700.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. A. Argyros and R. G. Haydon, A hereditarily indecomposable L ∞ -space that solves the scalar-plus-compact problem, Acta Mathematica 206 (2011), 1–54.

    Article  MathSciNet  MATH  Google Scholar 

  4. B. Bossard, A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fundamenta Mathematicae 172 (2002), 117–152.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bourgain, New Classes of Lp-spaces, Lecture Notes in Mathematics, Vol. 889, Springer, Berlin–New York, 1981.

  6. J. Bourgain and F. Delbaen, A class of special L ∞ spaces, Acta Mathematica 145 (1980), 155–176.

    Article  MathSciNet  MATH  Google Scholar 

  7. B. M. Braga, On the complexity of some inevitable classes of separable Banach spaces, Journal of Mathematical Analysis and Applications 431 (2015), 682–701.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Dodos, Banach Spaces and Descriptive Set Theory: Selected Topics, Lecture Notes in Mathematics, Vol. 1993, Springer, Berlin, 2010.

  9. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-dimensional Geometry, CMS Books in Mathematics, Vol. 8, Springer, New York, 2001.

    Google Scholar 

  10. T. Figiel and W. B. Johnson, A uniformly convex Banach space which contains no ℓp, Compositio Mathematica 29 (1974), 179–190.

    MathSciNet  MATH  Google Scholar 

  11. V. P. Fonf, One property of Lindenstrauss-Phelps spaces, Functional Analysis and its Applications 13 (1979), 66–67.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Ghawadrah, Non-isomorphic complemented subspaces of the reflexive Orlicz function spaces LF[0, 1], Proceedings of the American Mathematical Society 144 (2016), 285–299.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Ghawadrah, The descriptive complexity of the family of Banach spaces with the bounded approximation property, Houston Journal of Mathematics 43 (2017), 395–401.

    MathSciNet  MATH  Google Scholar 

  14. G. Godefroy, Analytic sets of Banach spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 104 (2010), 365–374.

    Google Scholar 

  15. G. Godefroy, The complexity of the isomorphism class of some Banach spaces, Journal of Nonlinear and Convex Analysis 18 (2017), 231–240.

    MathSciNet  Google Scholar 

  16. G. Godefroy, The isomorphism classes of ℓp are Borel, Houston Journal of Mathematics 43 (2017), 947–951.

    MathSciNet  MATH  Google Scholar 

  17. G. Godefroy, N. J. Kalton and G. Lancien, Szlenk indices and uniform homeomorphisms, Transactions of the American Mathematical Society 353 (2001), 3895–3918.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Godefroy and J. Saint-Raymond, Descriptive complexity of some isomorphic classes of Banach spaces, Journal of Functional Analysis 275 (2018), 1008–1022.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. B. Johnson and M. Zippin, On subspaces of quotients of \((\Sigma{G_n})_{{\ell}_{p}}\) and \((\Sigma{G_n})_{c_0}\), Israel Journal of Mathematics 13 (1972), 311–316.

    Article  MathSciNet  Google Scholar 

  20. A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts inMathematics, Vol. 156, Springer, New York, 1995.

  21. O. Kurka, Tsirelson-like spaces and complexity of classes of Banach spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matatemáticas 112 (2018), 1101–1123.

    MATH  Google Scholar 

  22. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer, Berlin–New York, 1977.

  23. V. Pták, A combinatorial theorem on systems of inequalities and its application to analysis, Czechoslovak Mathematical Journal 9 (1959), 629–630.

    MathSciNet  MATH  Google Scholar 

  24. H. P. Rosenthal, A characterization of c0 and some remarks concerning the Grothendieck property, in Texas Functional Analysis Seminar 1982–1983 (Austin, TX), Longhorn Notes, University of Texas Press, Austin, TX, 1983, pp. 95–108.

    Google Scholar 

  25. B. S. Tsirelson, Not every Banach space contains an imbedding of ℓp or c 0, Functional Analysis and its Applications 8 (1974), 138–141.

    Article  Google Scholar 

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Acknowledgment

The author is grateful to Gilles Godefroy for useful remarks on the topic and for hospitality during a visit at the Institut de Mathématiques de Jussieu. The author thanks Valentin Ferenczi and the anonymous referee for suggestions that helped to improve this work.

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Authors and Affiliations

  1. Mathematical Institute of Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Ondřej Kurka

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  1. Ondřej Kurka
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Correspondence to Ondřej Kurka.

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The author was supported by grant GAČR 17-00941S and by RVO: 67985840.

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Kurka, O. The isomorphism class of C0 is not Borel. Isr. J. Math. 231, 243–268 (2019). https://doi.org/10.1007/s11856-019-1851-0

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  • DOI: https://doi.org/10.1007/s11856-019-1851-0