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Unconditional families in Banach spaces

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Abstract

It is shown that for every separable Banach space X with non-separable dual, the space \(X^{**}\) contains an unconditional family of size \(|X^{**}|\) . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.

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References

  1. Apatsidis, D., Argyros, S.A., Kanellopoulos, V.: On the subspaces of JF and JT with non-separable dual (preprint)

  2. Argyros, S.A., Arvanitakis, A.D., Tolias, A.G.: Saturated extensions, the attractors method and hereditarily James tree spaces. In: Castillo, J.M., Johnson, W.B. (eds.) Methods in Banach spaces. LMS Lecture Notes, 337, pp. 1–90

  3. Argyros, S.A., Dodos, P., Kanellopoulos, V.: A classification of separable Rosenthal compacta and its applications. Dissertations Math (to appear)

  4. Argyros S.A., Felouzis V., Kanellopoulos V. (2002). A proof of Halpern-Läuchli Partition Theorem. Eur. J. Comb. 23: 1–10

    Article  MATH  MathSciNet  Google Scholar 

  5. Argyros, S.A., Godefroy, G., Rosenthal, H.P.: Descriptive set theory and Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2. Elsevier, Amsterdam (2003)

  6. Argyros, S.A., Todorčević, S.: Ramsey methods in analysis. Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, Verlag, Basel (2005)

  7. Argyros, S.A., Tolias, A.: Methods in the theory of hereditarily indecomposable Banach spaces. Memoirs AMS 806 (2004)

  8. Bessaga C., Pełczyński A. (1958). On bases and unconditional convergence of series in Banach spaces. Stud. Math. 17: 151–164

    MATH  Google Scholar 

  9. Blass A. (1981). A partition for perfect sets. Proc. AMS 82: 271–277

    Article  MATH  MathSciNet  Google Scholar 

  10. Carlson T.J. (1988). Some unifying principles in Ramsey theory. Discrete Math. 68: 117–169

    Article  MathSciNet  Google Scholar 

  11. Edgar G.A., Wheeler R.F. (1984). Topological properties of Banach spaces. Pacific J. Math. 115: 317–350

    MATH  MathSciNet  Google Scholar 

  12. Ellentuck E. (1974). A new proof that analytic sets are Ramsey. J. Symb. Logic 39: 163–165

    Article  MATH  MathSciNet  Google Scholar 

  13. Farahat, J.: Espaces be Banach contenant \(\ell_1\) d’ apres H.P. Rosenthal. Seminaire Maurey-Schwartz, Ecole Polytechnique, pp. 1973–74

  14. Godefroy G., Louveau A. (1989). Axioms of determinacy and biorthogonal systems. Israel J. Math. 67: 109–116

    Article  MATH  MathSciNet  Google Scholar 

  15. Godefroy G., Talagrand M. (1982). Espaces de Banach représentable. Israel J. Math. 41: 321–330

    Article  MATH  MathSciNet  Google Scholar 

  16. Gowers W.T. (2002). An infinite Ramsey theorem and some Banach-space dichotomies. Ann. Math. 156: 797–833

    Article  MATH  MathSciNet  Google Scholar 

  17. Gowers, W.T.: Ramsey methods in Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2. Elsevier, Amsterdam (2003)

  18. Gowers W.T., Maurey B. (1993). The unconditional basic sequence problem. J. AMS 6: 851–874

    MATH  MathSciNet  Google Scholar 

  19. Hagler J.N., Johnson W.B. (1977). On Banach spaces whose dual balls are not weak* sequentially compact. Israel J. Math. 28: 325–330

    Article  MATH  MathSciNet  Google Scholar 

  20. Halpern J.D., Läuchli H. (1966). A partition theorem. Trans. AMS 124: 360–367

    Article  MATH  Google Scholar 

  21. James R.C. (1974). A separable somewhat reflexive Banach space with non-separable dual. Bull. AMS 80: 738–743

    MATH  Google Scholar 

  22. Johnson W.B., Rosenthal H.P. (1972). On weak* basic sequences and their applications to the study of Banach spaces. Stud. Math. 43: 77–92

    MathSciNet  Google Scholar 

  23. Kanellopoulos V. (2005). Ramsey families of subtrees of the dyadic tree. Trans. AMS 357: 3865–3886

    Article  MATH  MathSciNet  Google Scholar 

  24. Kechris, A.S.: Classical Descriptive Set Theory. Grad. Texts in Math. 156, Springer, Heidelberg (1995)

  25. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces I and II. Springer, Heidelberg (1996)

  26. Louveau A., Shelah S., Veličković B. (1993). Borel partitions of infinite subtrees of a perfect tree. Ann. Pure Appl. Logic 63: 271–281

    Article  MATH  MathSciNet  Google Scholar 

  27. Maurey, B.: Types and \(\ell_1\) subspaces. Texas functional analysis seminar 1982–1983 (Austin, Tex.), 123-137, Longhorn Notes, University of Texas Press, Austin (1983)

  28. Milliken K. (1981). A partition theorem for the infinite subtrees of a tree. Trans. AMS 263: 137–148

    Article  MATH  MathSciNet  Google Scholar 

  29. Miller A.W. (1989). Infinite combinatorics and definability. Ann. Pure Appl. Logic 41: 179–203

    Article  MATH  MathSciNet  Google Scholar 

  30. Milman V.D. (1970). Geometric theory of Banach spaces. I. Theory of basic and minimal systems (Russian). Uspehi Mat. Nauk 25: 113–174

    MathSciNet  Google Scholar 

  31. Odell, E.: Applications of Ramsey theorems to Banach space theory. Notes in Banach spaces, 379–404, University of Texas Press, Austin (1980)

  32. Odell E., Rosenthal H.P. (1975). A double-dual characterization of separable Banach spaces not containing \(\ell_1\). Israel J. Math. 20: 375–384

    Article  MATH  MathSciNet  Google Scholar 

  33. Pawlikowski J. (1990). Parametrized ellentuck theorem. Topol. Appl. 37: 65–73

    Article  MATH  MathSciNet  Google Scholar 

  34. Pełczyński A. (1968). On Banach spaces containing \(L_1(\mu)\). Stud. Math. 30: 231–246

    Google Scholar 

  35. Rosenthal H.P. (1974). A characterization of Banach spaces containing \(\ell_1\). Proc. Nat. Acad. Sci. USA 71: 2411–2413

    Article  MATH  Google Scholar 

  36. Stegall C. (1975). The Radon-Nikodym property in conjugate Banach spaces. Trans. AMS 206: 213–223

    Article  MATH  MathSciNet  Google Scholar 

  37. Stern J. (1978). A Ramsey theorem for trees with an application to Banach spaces. Israel J. Math. 29: 179–188

    Article  MATH  MathSciNet  Google Scholar 

  38. Todorčević S. (1999). Compact subsets of the first Baire class. J. AMS 12: 1179–1212

    Google Scholar 

  39. Todorčević, S.: Introduction to Ramsey spaces (to appear)

  40. Todorčević S. (2006). Biorthogonal systems and quotient spaces via Baire category methods. Math. Ann. 335: 687–715

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Applied Sciences, Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece

    S. A. Argyros, P. Dodos & V. Kanellopoulos

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  1. S. A. Argyros
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  2. P. Dodos
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  3. V. Kanellopoulos
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Correspondence to S. A. Argyros.

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Argyros, S.A., Dodos, P. & Kanellopoulos, V. Unconditional families in Banach spaces. Math. Ann. 341, 15–38 (2008). https://doi.org/10.1007/s00208-007-0179-y

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