, Volume 10, Issue 2, pp 365–390 | Cite as

The Radon–Nikodym Property for Tensor Products of Banach Lattices



Let 1 ≤ p < ∞. We show that Open image in new window , the Fremlin projective tensor product of p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that Open image in new window , the Wittstock injective tensor product of p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from p' to X is compact, where 1/p +1/p'= 1 and let p' = c0 if p = 1.

Mathematics Subject Classification 2000

46B22 46B42 46M05 47B65 


Radon–Nikodym property operators on Banach lattices injective tensor product projective tensor product sequence spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramovich, Y.A. and Aliprantis, C.D.: An Introduction to Operator Theory, Graduate Studies in Mathematics, Volume 50, AMS, 2002.Google Scholar
  2. 2.
    Abramovich, Y.A., Chen, Z.L. and Wickstead, A.W.: Regular-norm balls can be closed in the strong operator topology, Positivity 1 (1997), 75–96.Google Scholar
  3. 3.
    Bourgain, J. and Rosenthal, H.P.: Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), 149–188.Google Scholar
  4. 4.
    Bu, Q.: Some properties of the injective tensor product of Lp[0,1] and a Banach space, J. Funct. Anal. 204 (2003), 101–121.Google Scholar
  5. 5.
    Bu, Q. and Diestel, J.: Observations about the projective tensor product of Banach spaces, I –- Open image in new window Quaestiones Math. 24 (2001), 519–533.Google Scholar
  6. 6.
    Buskes, G. and van Rooij, A.: Bounded variation and tensor products of Banach lattices, Positivity 7 (2003), 47–59.Google Scholar
  7. 7.
    Cartwright, D.I. and Lotz, H.P.: Some characterizations of AM- and AL-spaces, Math. Z. 142 (1975), 97–103.Google Scholar
  8. 8.
    Chen, Z.L. and Wickstead, A.W.: Some applications of Rademacher sequences in Banach lattices, Positivity 2 (1998), 171–191.Google Scholar
  9. 9.
    Defant, A. and Floret, K.: Tensor Norms and Operator Ideals, Noth-Holland, Amsterdam, 1993.Google Scholar
  10. 10.
    Diestel, J., Jarchow, H. and Tonge, A.: Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.Google Scholar
  11. 11.
    Diestel, J. and Uhl, J.J. Vector measures, Math. Surveys Vol. 15, American Mathematical Society, Providence, RI, 1977.Google Scholar
  12. 12.
    Fourie, J.H. and Röntgen, I.M.: Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2003), 629–644.Google Scholar
  13. 13.
    Fremlin, D.H.: Tensor products of Archimedean vector lattices, Am. J. Math. 94 (1972), 778–798.Google Scholar
  14. 14.
    Fremlin, D.H.: Tensor products of Banach lattices, Math. Ann. 211 (1974), 87–106.Google Scholar
  15. 15.
    Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires, Mem. Am. Math. Soc. 16 (1955).Google Scholar
  16. 16.
    Leonard, I.E.: Banach sequence spaces, J. Math. Anal. Appl. 54 (1976), 245–265.Google Scholar
  17. 17.
    Lotz, H.P., Peck, N.T. and Porta, H.: Semi-embedding of Banach spaces, Proc. Edinburgh Math. Soc. 22 (1979), 233–240.Google Scholar
  18. 18.
    Meyer-Nieberg, P.: Banach Lattices, Springer, 1991.Google Scholar
  19. 19.
    Ryan, R.A.: Introduction to Tensor Products of Banach Spaces, Springer, 2002.Google Scholar
  20. 20.
    Wittstock, G.: Eine Bemerkung über Tensorprodukte von Banachverbänden, Arch. Math. 25 (1974), 627–634.Google Scholar
  21. 21.
    Wittstock, G.: Ordered normed tensor products, Lecture Notes in Physics, Vol. 29, Springer, 1974, pp. 67–84.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

Personalised recommendations