Mathematische Annalen

, Volume 367, Issue 3–4, pp 1777–1790 | Cite as

Shellable weakly compact subsets of C[0, 1]

Article
  • 106 Downloads

Abstract

We show that for every weakly compact subset K of C[0, 1] with finite Cantor–Bendixson rank, there is a reflexive Banach lattice E and an operator \(T:E\rightarrow C[0,1]\) such that \(K\subseteq T(B_E)\). On the other hand, we exhibit an example of a weakly compact set of C[0, 1] homeomorphic to \(\omega ^\omega +1\) for which such T and E cannot exist. This answers a question of M. Talagrand in the 80’s.

Mathematics Subject Classification

46B50 46B42 47B07 

References

  1. 1.
    Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Springer, New York (2006)MATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Factoring compact and weakly compact operators through reflexive Banach lattices. Trans. Am. Math. Soc. 283, 369–381 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Reprint of the 1985 original. Springer, Dordrecht (2006)Google Scholar
  4. 4.
    Alspach, D.E.: Quotients of \(C[0,1]\) with separable dual. Isr. J. Math. 29(4), 361–384 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Argyros, S.A., Todorcevic, S.: Ramsey Methods in Analysis. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2005)MATHGoogle Scholar
  6. 6.
    Blanco, A., Kaijser, S., Ransford, T.J.: Real interpolation of Banach algebras and factorization of weakly compact homomorphisms. J. Funct. Anal. 217(1), 126–141 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, Z.L., Wickstead, A.W.: Relative weak compactness of solid hulls in Banach lattices. Indag. Math. (N.S.) 9(2), 187–196 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Davis, W.J., Figiel, T., Johnson, W.B., Pełczyński, A.: Factoring weakly compact operators. J. Funct. Anal. 17, 311–327 (1974)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Flores, J., Tradacete, P.: Factorization and domination of positive Banach-Saks operators. Studia Math. 189(1), 91–101 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gasparis, I.: Operators on \(C[0,1]\) preserving copies of asymptotic \(\ell _1\) spaces. Math. Ann. 333(4), 831–858 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, vol. 92. Springer, Berlin, New York (1977)CrossRefMATHGoogle Scholar
  12. 12.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, New York (1979)CrossRefMATHGoogle Scholar
  13. 13.
    López-Abad, J., Ruiz, C., Tradacete, P.: The convex hull of a Banach-Saks set. J. Funct. Anal. 266(4), 2251–2280 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mazurkiewicz, S., Sierpiński, W.: Contribution à la topologie des ensembles dénombrables. Fundamenta Mathematicae 1, 17–27 (1920)MATHGoogle Scholar
  15. 15.
    Rosenthal, H.P.: The Banach Spaces C(K). Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1547–1602. North-Holland, Amsterdam (2003)Google Scholar
  16. 16.
    Szlenk, W.: Sur les suites faiblement convergentes dans l’espace \(L\). Studia Math. 25, 337–341 (1965)MathSciNetMATHGoogle Scholar
  17. 17.
    Talagrand, M.: Some weakly compact operators between Banach lattices do not factor through reflexive Banach lattices. Proc. Am. Math. Soc. 96(1), 95–102 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Instituto de Ciencias Matematicas (ICMAT)CSIC-UAM-UC3M-UCMMadridSpain
  2. 2.Mathematics DepartmentUniversidad Carlos III de MadridLeganés (Madrid)Spain

Personalised recommendations