Skip to main content
SpringerLink
Log in

A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

  • Published:
Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society Aims and scope Submit manuscript

Abstract

Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ λ∞ -spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ2, we obtain a ℒ λ∞ -spaceX with the RNP such that the projective tensor product\(X\hat \otimes X\) containsc 0 and hence fails the RNP. TakingE=L 1, we obtain a ℒ λ∞ -space failing the RNP but nevertheless not containingc 0.

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. J. Bourgain; New classes of ℒp spaces Springer Lecture Notes n.o 899.

  2. J. Bourgain—F. Delbaen; A class of special ℒ ∞ spaces, Acta Math. 145 (1980) 155–176.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Diestel, J. Uhl;Vector measures, Amer. Math. Soc. Math. Surveys 15 (1977) Providence-Rhode-Island.

  4. A. Grothendieck;Résumé de la théorie métrique ddes produits tensoriels topologiques, Bol. Soc. Mat. São-Paulo, 8 (1956) 1–79.

    Google Scholar 

  5. V.I. Gurarii;Space of universal disposition, Isotopic spaces and the Mazur problem on rotations of Banach spaces, Sibirsk. Math. Z. 7 (1966) 1002–1013 (Russian).

    MathSciNet  Google Scholar 

  6. S.V. Kisliakov;On spaces with “small” annihilators, Zap. Naucn. Sem. Leningrad, Otdel. Math. Inst. Steklov (LOMI) 65 (1976) 192–195.

    Google Scholar 

  7. G. Pisier;Counterexamples to a conjecture of Grothendieck, Acta Math., to appear.

  8. J. Bourgain and H.P. Rosenthal;Martingales values in certain subspaces of L 1, Israel J. Math. 27 (1980) 54–75.

    MathSciNet  Google Scholar 

  9. M. Talagrand;Sur la propriété de Radon-Nikodym dans les espaces de Banach reticulés, C.R. Acad. Sci. Paris. Série A (288) (1979) p. 907–910.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. Wiskunde, Vrije Universiteit Brussels, Pleinlaan 2, Brussels, Belgium

    Jean Bourgain

  2. Equipe D'Analyse, Université Paris 6, Tour 46 4, Place Jussieu, 75230, Paris Cedex 05

    Gilles Pisier

Authors
  1. Jean Bourgain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Pisier
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Bourgain, J., Pisier, G. A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces. Bol. Soc. Bras. Mat 14, 109–123 (1983). https://doi.org/10.1007/BF02584862

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02584862

Keywords

Search

Navigation