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The kadec-klee property in symmetric spaces of measurable operators

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Abstract

We show that ifE is a separable symmetric Banach function space on the positive half-line thenE has the Kadec-Klee property if and only if, for every semifinite von Neumann algebra (M, τ), the associated spaceE(M, τ) ofτ-measurable operators has the Kadec-Klee property.

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References

  1. J. Arazy,More on convergence in unitary matrix spaces, Proceedings of the American Mathematical Society83 (1981), 44–48.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Bennett and R. Sharpley,Interpolation of Operators, Academic Press, New York, 1988.

    MATH  Google Scholar 

  3. V. I. Chilin, P. G. Dodds, A. A. Sedaev and F. A. Sukochev,Characterization of Kadec-Klee properties in symmetric spaces of measurable functions, preprint, 1994.

  4. V. I. Chilin and F. A. Sukochev,Measure convergence in regular non-commutative symmetric spaces, Izv. VUZov (Matematika)9 (1990), 63–70 (Russian); English translation: Soviet Mathematics34 (1990), 78–87.

    MathSciNet  Google Scholar 

  5. V. I. Chilin and F. A. Sukochev,Weak convergence in non-commutative symmetric spaces, Journal of Operator Theory31 (1994), 35–65.

    MATH  MathSciNet  Google Scholar 

  6. P. G. Dodds, T. K. Dodds and B. de Pagter,Non-commutative Banach function spaces, Mathematische Zeitschrift201 (1989), 583–597.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. G. Dodds, T. K. Dodds and B. de Pagter,Fully symmetric operator spaces, Integral Equations and Operator Theory15 (1992), 942–972.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. G. Dodds, T. K. Dodds and B. de Pagter,Non-commutative Köthe duality, Transactions of the American Mathematical Society339 (1993), 717–750.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. J. Davis, N. Ghoussoub and J. Lindenstrauss,A lattice renorming theorem and applications to vector-valued processes, Transactions of the American Mathematical Society263 (1981), 531–540.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Fack and H. Kosaki,Generalized s-numbers of τ-measurable operators, Pacific Journal of Mathematics123 (1986), 269–300.

    MATH  MathSciNet  Google Scholar 

  11. M. I. Kadec and A. Pelczynski,Bases, lacunary sequences and complemented subspaces in the spaces L p , Studia Mathematica21 (1962), 161–176.

    MATH  MathSciNet  Google Scholar 

  12. S. G. Krein, Ju. I. Petunin and E. M. Semenov,Interpolation of linear operators, Translations of Mathematical Monographs, Vol. 54, American Mathematical Society, 1982.

  13. A. V. Krygin, F. A. Sukochev and V. E. Sheremetjev,Convergence in measure, weak convergence and structure of subspaces in symmetric spaces of measurable operators, Dep. VINITI, N2487-B92, 1-34 (Russian).

  14. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Springer-Verlag, Berlin, 1979.

    MATH  Google Scholar 

  15. E. Nelson,Notes on non-commutative integration, Journal of Functional Analysis15 (1974), 103–116.

    Article  MATH  Google Scholar 

  16. V. I. Ovčinnikov,s-numbers of measurable operators, Funktsional’nyi Analiz i Ego Prilozheniya4 (1970), 78–85 (Russian).

    Google Scholar 

  17. V. I. Ovčinnikov,Symmetric spaces of measurable operators, Dokl. Nauk SSSR191 (1970), 769–771 (Russian); English Translation: Soviet Math. Dokl.11 (1970), 448–451.

    Google Scholar 

  18. A. A. Sedaev,On the (H)-property in symmetric spaces, Teoriya funkcii, Func. Anal. i Prilozenia11 (1970), 67–80 (Russian).

    MATH  MathSciNet  Google Scholar 

  19. A. A. Sedaev,On weak and norm convergence in interpolation spaces, Trudy 6 zimney shkoly po mat. programm. i smezn. voprosam. Moscow, 1975, pp. 245-267 (Russian).

  20. B. Simon,Convergence in trace ideals, Proceedings of the American Mathematical Society83 (1981), 39–43.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. A. Sukochev,Non-isomorphism of L p -spaces associated with finite and infinite von Neumann algebras, Proceedings of the American Mathematical Society (to appear).

  22. S. Stratila and L. Zsido,Lectures on Von Neumann Algebras, Editura and Abacus Press, 1979.

  23. M. Takesaki,Theory of Operator Algebras I, Springer-Verlag, New York-Heidelberg-Berlin, 1979.

    MATH  Google Scholar 

  24. M. Terp,L p-spaces associated with von Neumann algebras, Notes, Copenhagen University, 1981.

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Authors and Affiliations

  1. Department of Mathematics, Tashkent State University, 700095, Tashkent, Uzbekistan

    V. I. Chilin

  2. Department of Mathematics and Statistics, The Flinders University of South Australia, GPO Box 2100, 5001, Adelaide, SA, Australia

    P. G. Dodds & F. A. Sukochev

Authors
  1. V. I. Chilin
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  2. P. G. Dodds
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  3. F. A. Sukochev
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Research supported by the Australian Research Council.

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Chilin, V.I., Dodds, P.G. & Sukochev, F.A. The kadec-klee property in symmetric spaces of measurable operators. Isr. J. Math. 97, 203–219 (1997). https://doi.org/10.1007/BF02774037

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  • DOI: https://doi.org/10.1007/BF02774037

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