, Volume 15, Issue 4, pp 571–594 | Cite as

Properties (u) and (V*) of Pelczynski in symmetric spaces of τ-measurable operators

  • P. G. DoddsEmail author
  • B. de Pagter
Open Access


It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V *) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.


Measurable operators Property (u) Property (V*) 

Mathematics Subject Classification (2000)

Primary 46L52 Secondary 46E30 47A30 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, London (1985)Google Scholar
  2. 2.
    Bombal F.: (V *) sets and Pelczynski’s property (V *). Glasg. Math. J. 32, 109–120 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Diestel, J., Uhl, J.J. Jr.: Vector Measures. In: Math. Surveys, vol. 15. American Mathematical Society, Providence (1977)Google Scholar
  4. 4.
    Diestel, J.: Sequences and series in Banach spaces. In: Graduate Texts in Mathematics, vol. 92. Springer, Berlin (1984)Google Scholar
  5. 5.
    Dixmier, J.: von Neumann Algebras. In: Mathematical Library, vol. 27. North Holland, Amsterdam (1981)Google Scholar
  6. 6.
    Dodds P.G., Dodds T.K., de Pagter B.: Non-commutative Banach function spaces. Math. Z. 201, 583–597 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dodds P.G., Dodds T.K., de Pagter B.: Weakly compact subsets of symmetric operator spaces. Math. Proc. Camb. Philos. Soc. 110, 169–182 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dodds P.G., Dodds T.K., de Pagter B.: Non-commutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dodds, P.G., de Pagter, B., The non-commutative Yosida–Hewitt decomposition revisited. Trans. Am. Math. Soc (in press)Google Scholar
  10. 10.
    Fack T., Kosaki H.: Generalized s-numbers of τ-measurable operators. Pac. J. Math. 123, 269–300 (1986)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Krein, S.G., Petunin, Ju.I., Semenov, E.M.: Interpolation of linear operators. In: Translations of Mathematical Monographs, vol. 54. American Mathematical Society, Providence (1982)Google Scholar
  12. 12.
    Lozanovskiĭ G.Ya.: On isomorphic Banach lattices. Sib. Math. J. 10, 67–71 (1969)Google Scholar
  13. 13.
    Luxemburg, W.A.J.: Banach Function Spaces, PhD Thesis, Delft Institute of Technology (1955)Google Scholar
  14. 14.
    Luxemburg W.A.J.: Notes on Banach Function Spaces, XIVA. Indag. Math. 27, 229–239 (1965)MathSciNetGoogle Scholar
  15. 15.
    Meyer-Nieberg P.: Zur schwachen Kompactheit in Banachverbänden. Math. Z. 134, 303–315 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)Google Scholar
  17. 17.
    Nelson E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)zbMATHCrossRefGoogle Scholar
  18. 18.
    Ogasawara, T.: Vector Lattices. Tokyo (1948) (in Japanese)Google Scholar
  19. 19.
    Pelczyński A.: A connection between weakly unconditional convergence and weak completeness of Banach spaces. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 6, 251–253 (1958)zbMATHGoogle Scholar
  20. 20.
    Pelczyński A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Pol. Sci. 10, 641–648 (1962)zbMATHGoogle Scholar
  21. 21.
    Pfitzner H.: l-summands in their biduals have Pelczynski’s property (V *). Studia Math. 104, 91–98 (1994)MathSciNetGoogle Scholar
  22. 22.
    Randrianantoanina N.: Pelczynski’s property (V *) for symmetric operator spaces. Proc. Am. Math. Soc. 125, 801–806 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Saab E., Saab P.: On Pelczynski’s properties (V) and (V *). Pac. J. Math. 125, 205–210 (1986)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Takesaki M.: Theory of Operator Algebras I. Springer, New-York (1979)zbMATHCrossRefGoogle Scholar
  25. 25.
    Terp, M.: L p-spaces associated with von Neumann algebras. In: Notes. Copenhagen University, Copenhagen (1981)Google Scholar
  26. 26.
    Tzafriri L.: Reflexivity in Banach lattices and their subspaces. J. Funct. Anal. 10, 1–18 (1972)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.School of Computer Science, Mathematics and EngineeringFlinders UniversityAdelaideAustralia
  2. 2.Delft Institute of Applied Mathematics, Faculty EEMCSDelft University of TechnologyDelftThe Netherlands

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