Skip to main content
SpringerLink
Log in

The universality of ℓ 1 as a dual space

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a \({{\mathcal L}_\infty}\) space Z whose dual is isomorphic to ℓ 1. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alspach, D.: A l 1-predual which is not isometric to a quotient of C(α). In: Banach Spaces (Mérida, 1992). Contemp. Math., vol. 144. Am. Math. Soc., pp. 9–14 (1993)

  2. Alspach D.: The dual of the Bourgain–Delbaen space. Israel J. Math. 117, 239–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alspach D., Judd R., Odell E.: The Szlenk index and local ℓ 1-indices. Positivity 9(1), 1–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Argyros, S.A., Haydon, R.: A hereditarily indecomposable \({{\mathcal L}_\infty}\) -space that solves the scalar- plus-compact (preprint). arXiv:0903.3921

  5. Bourgain J.: On convergent sequences of continuous functions. Bull. Soc. Math. Belg. Sér. B 32(2), 235–249 (1980)

    MathSciNet  MATH  Google Scholar 

  6. Bourgain J.: The Szlenk index and operators on C(K)-spaces. Bull. Soc. Math. Belg. Sér. B 31(1), 87–117 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Bourgain J., Delbaen F.: A class of special \({{\mathcal L}_\infty}\) spaces. Acta Math. 145(3-4), 155–176 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourgain J., Pisier G.: A construction of \({{\mathcal L}_{\infty} }\) -spaces and related Banach spaces. Bol. Soc. Brasil. Mat. 14(2), 109–123 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casazza P.G., Johnson W.B., Tzafriri L.: On Tsirelson space. Israel J. Math. 47(2-3), 81–98 (1984)

    Article  MathSciNet  Google Scholar 

  10. Dodos P.: On classes of Banach spaces admitting small universal spaces. Trans. A.M.S. 361, 6407–6428 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fonf V.: A property of Lindenstrauss Phelps spaces. Funct. Anal. Appl. 13(1), 66–67 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Freeman D., Odell E., Schlumprecht Th., Zsák A.: Banach spaces of bounded Szlenk index, II. Fund. Math. 205(2), 161–177 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Godefroy, G.: The Szlenk index and its applications. General topology in Banach spaces, pp. 71–79. Nova Sci. Publ., Huntington, NY (2001)

  14. Godefroy G., Kalton N.J., Lancien G.: Szlenk indices and uniform homeomorphisms. Trans. Am. Math. Soc. 353(10), 3895–3918 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Haydon R.: Subspaces of the Bourgain–Delbaen space. Studia Math. 139(3), 275–293 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. I, pp. 1–84. North-Holland, Amsterdam (2001)

  17. Johnson W.B., Rosenthal H.P., Zippin M.: On bases, finite dimensional decompositions, and weaker structures in Banach spaces. Israel J. Math. 9, 488–506 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Johnson W.B., Zippin M.: Separable L 1 preduals are quotients of C(Δ). Israel J. Math. 16, 198–202 (1973)

    Article  MathSciNet  Google Scholar 

  19. Johnson W.B., Zippin M.: Subspaces and quotient spaces of \({(\sum G_n)_{\ell_p}}\) and \({(\sum G_n)_{c_0}}\) . Israel J. Math. 17, 50–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Judd R., Odell E.: Concerning Bourgain’s ℓ 1 index of a Banach space. Israel J. Math. 108, 145–171 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lancien G.: A survey on the Szlenk index and some of its applications. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100(1-2), 209–235 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Leung D.H., Tang W.-K.: The Bourgain ℓ 1-index of mixed Tsirelson space. J. Funct. Anal. 199(2), 301–331 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces, I. In: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin/New York (1977)

  24. Odell E., Schlumprecht Th.: Trees and branches in Banach spaces. Trans. Am. Math. Soc. 354(10), 4085–4108 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Odell, E., Schlumprecht, Th., Zsák, A.: A new infinite game in Banach spaces with applications. In: Banach Spaces and Their Applications in Analysis, pp. 147–182. Walter de Gruyter, Berlin (2007)

  26. Odell E., Schlumprecht Th., Zsák A.: Banach spaces of bounded Szlenk index. Studia Math. 183(1), 63–97 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Odell E., Tomczak-Jaegermann N., Wagner R.: Proximity to ℓ 1 and distortion in asymptotic ℓ 1 spaces. J. Funct. Anal. 150(1), 101–145 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Szlenk W.: The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces. Studia Math. 30, 53–61 (1968)

    MathSciNet  MATH  Google Scholar 

  29. Zippin M.: On some subspaces of Banach spaces whose duals are L 1 spaces. Proc. Am. Math. Soc. 23, 378–385 (1969)

    MathSciNet  MATH  Google Scholar 

  30. Zippin M.: Banach spaces with separable duals. Trans. Am. Math. Soc. 310(1), 371–379 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    D. Freeman & Th. Schlumprecht

  2. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-0257, USA

    E. Odell

Authors
  1. D. Freeman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Odell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Th. Schlumprecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Th. Schlumprecht.

Additional information

E. Odell and Th. Schlumprecht was supported by the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Freeman, D., Odell, E. & Schlumprecht, T. The universality of ℓ 1 as a dual space. Math. Ann. 351, 149–186 (2011). https://doi.org/10.1007/s00208-010-0601-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0601-8

Mathematics Subject Classification (2000)

Search

Navigation