Some Properties of the Injective Tensor Product of Banach Spaces

ORIGINAL ARTICLES
  • 49 Downloads

Abstract

Let X and Y be Banach spaces such that X has an unconditional basis. Then X\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } \)Y , the injective tensor product of X and Y , has the Radon–Nikodym property (respectively, the analytic Radon–Nikodym property, the near Radon–Nikodym property, non-containment of a copy of c0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.

Keywords

tensor products types of Radon–Nikodym properties Schauder decomposition 

MR (2000) Subject Classification

46M05 46B28 

References

  1. 1.
    Diestel, J., Uhl, J. J.: Vector Measures, Mathematical Surveys, vol. 15, American Mathematical Society, Providence, R.I., 1977Google Scholar
  2. 2.
    Bourgain, J., Pisier, G.: A construction of ℒ-spaces and related Banach spaces. Bol. Soc. Brasil. Mat., 14, 109–123 (1983)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Diestel, J., Uhl J. J.: The Radon-Nikodym theorem for Banach space valued measures. Rocky Mountain J. Math., 6, 1–46 (1976)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Andrews, K. T.: The Radon–Nikodym property for spaces of operators. J. London Math. Soc., 28, 113–122 (1983)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bu, Q., Lin, P. K.: Radon–Nikodym property for the projective tensor product of Köthe function spaces. J. Math. Anal. Appl., 293, 149–159 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bu, Q., Dowling, P. N.: Observations about the projective tensor product of Banach spaces, III — L p[0, 1]⊗X, 1 < p < ∞. Quaestiones Math., 25, 303–310 (2002)MATHMathSciNetGoogle Scholar
  7. 7.
    Bu, Q., Diestel, J., Dowling, P. N., Oja, E.: Types of Radon-Nikodym properties for the projective tensor product of Banach spaces. Illinois J. Math., 47, 1303–1326 (2003)MATHMathSciNetGoogle Scholar
  8. 8.
    Diestel, J., Morrison, T. J.: The Radon-Nikodym property for the space of operators, I. Math. Nachr., 92, 7–12 (1979)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Randrianantoanina, N.: Radon-Nikodym properties for spaces of compact operators. Rev. Roumaine Math. Pures Appl., 41, 119–131 (1996)MATHMathSciNetGoogle Scholar
  10. 10.
    Edgar, G. A.: Banach spaces with the analytic Radon-Nikodym property and compact abelian groups, Almost everywhere convergence (Columbus, OH, 1988), Academic Press, Boston, MA, pp. 195–213 (1989)Google Scholar
  11. 11.
    Dowling, P. N.: Radon-Nikodym properties associated with subsets of countable discrete abelian groups. Trans. Amer. Math. Soc., 327, 879–890 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rudin, W.: Fourier analysis on groups, Interscience Publishers, New York–London, 1962Google Scholar
  13. 13.
    Dowling, P. N.: Duality in some vector-valued function spaces. Rocky Mountain J. Math., 22, 511–518 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaufman, R., Petrakis, M., Riddle, L., Uhl, J. J.: Nearly representable operators. Trans. Amer. Math. Soc., 312, 315–333 (1989)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, 1977Google Scholar
  16. 16.
    Lin, P. K.: Köthe–Bochner Function Spaces, Birkhäuser Boston, Inc., Boston, MA, 2004Google Scholar
  17. 17.
    Kalton, N.: Schauder decompositions and completeness. Bull. London Math. Soc., 2, 34–36 (1970)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Bu, Q.: Semi-embeddings and weakly sequential completeness of the projective tensor product. Studia Math., 169, 287–294 (2005)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lewis, D. R.: Conditional weak compactness in certain inductive tensor products. Math. Ann., 201, 201–209 (1973)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinP. R. China
  2. 2.Department of MathematicsSun Yat-sen UniversityGuangzhouP. R. China
  3. 3.Department of MathematicsUniversity of MississippiUniversityUSA

Personalised recommendations