, Volume 9, Issue 3, pp 457–484 | Cite as

A Non-commutative Yosida–Hewitt Theorem and Convex Sets of Measurable Operators Closed Locally in Measure

  • P. G. DoddsEmail author
  • T. K. Dodds
  • F. A. Sukochev
  • O. Ye. Tikhonov


We present a non-commutative extension of the classical Yosida–Hewitt decomposition of a finitely additive measure into its σ-additive and singular parts. Several applications are given to the characterisation of bounded convex sets in Banach spaces of measurable operators which are closed locally in measure.


non-commutative Banach function spaces singular functionals measurable operators local convergence in measure Köthe duality 


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  1. 1.
    Aliprantis, C. D., Burkinshaw, O. 1985Positive OperatorsAcademic Press Inc.New YorkGoogle Scholar
  2. 2.
    Ando, T. 1962On fundamental properties of a Banach space with a conePacific J. Math.1211631169Google Scholar
  3. 3.
    Bikchentaev, A.M. 2004The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebraLobachevskii J. Math.141724Google Scholar
  4. 4.
    Bukhvalov, A.V. and Ja. Lozanovskii, G.: On sets closed with respect to convergence in measure in spaces of measurable functions, Dokl. Akad. Nauk SSSR 212 (1973), 1273–1275; English transl. in Soviet Math. Doklady 14 (1973).Google Scholar
  5. 5.
    Bukhvalov, A.V. and Ja. Lozanovskii, G.: On sets closed in measure in spaces of measurable functions, Trudy Mosk. Matem. ob-va 34(1977), 129–150; English transl. in Trans. Moscow Math. Soc. (2) (1978), 127–148.Google Scholar
  6. 6.
    Bennett, C., Sharpley, R. 1988Interpolation of OperatorsAcademic PressNew YorkGoogle Scholar
  7. 7.
    Bukhvalov, A.V., Veksler, A.I., Ja. Lozanovskil, G. 1979Banach lattices-some Banach aspects of their theoryRussian Math. Surveys34159212Google Scholar
  8. 8.
    Bukhvalov, A.V. 1995Optimization without compactness and its applicationsOperator Theory Adv. Appl.7595112Google Scholar
  9. 9.
    Dodds, P.G., Dodds, T.K., Pagter, B. 1989Non-commutative Banach function spacesMath. Z.201583597CrossRefGoogle Scholar
  10. 10.
    Dodds, P.G., Dodds, T.K., Pagter, B. 1992Fully symmetric operator spacesIntegr. Equat. Oper. Th.15942972CrossRefGoogle Scholar
  11. 11.
    Dodds, P.G., Dodds, T.K., Pagter, B. 1993Non-commutative Köthe dualityTrans. Am. Math. Soc.339717750Google Scholar
  12. 12.
    Dodds, P.G., Sukochev, F.A., Schlüctermann, G. 2001Weak compactness criteria in symmetric spaces of measurable operatorsMath. Proc. Camb. Phil. Soc.131363384CrossRefGoogle Scholar
  13. 13.
    Fack, T., Kosaki, H. 1986Generalized s-numbers of τ-measurable operatorsPacific J. Math.123269300Google Scholar
  14. 14.
    Kantorovich, L.V., Akilov, G.P. 1984Functional AnalysisNaukaMoscowGoogle Scholar
  15. 15.
    Krein, S.G., Petunin, Ju.I. and Semenov, E.M.: Interpolation of linear operators, Translations of Mathematical Monographs, Am. Math. Soc. 54 (1982).Google Scholar
  16. 16.
    Lindenstrauss, J., Tzafriri, L. 1979Classical Banach Spaces IISpringer-VerlagBerlinGoogle Scholar
  17. 17.
    Muratov, M.A. 1978Convergence in the ring of measurable operatorsCollection of scientific papers of Tashkent University5735158RussianGoogle Scholar
  18. 18.
    Nelson, E. 1974Notes on non-commutative integrationJ. Funct. Anal.15103116CrossRefGoogle Scholar
  19. 19.
    Sedaev, A.A. 1974On a problem of G. Ya. Lozanovskii, Trudy Nauchko-IssledInst. Mekhanika Voronezh Univ.146367Google Scholar
  20. 20.
    Skvortsova, G.Sh., Tikhonov, O.Ye. 1998Convex sets in noncommutative L1-spaces closed in the topology of local convergence in measureRussian Math. (Iz. VUZ)424652Google Scholar
  21. 21.
    Stratila, S. and Zsidó, L.: Lectures on von Neumann algebras, Editura and Abacus Press, 1979.Google Scholar
  22. 22.
    Takesaki, M. 1958On the conjugate space of operator algebraTôhoku Math. J.10194203Google Scholar
  23. 23.
    Takesaki, M. 1979Theory of Operator Algebras ISpringer-VerlagNew York-Heidelberg-BerlinGoogle Scholar
  24. 24.
    Takesaki, M. 2003Theory of Operator Algebras II, Encyclopaedia of Mathematical SciencesSpringer-VerlagNew York-Heidelberg-BerlinVol. 125Google Scholar
  25. 25.
    Terp, M.: Lp-spaces Associated with von Neumann Algebras, Notes, Copenhagen Univ. (1981).Google Scholar
  26. 26.
    Yosida, K., Hewitt, E. 1952Finitely additive measuresTrans. Am. Math. Soc.724666Google Scholar
  27. 27.
    Zaanen, A.C. 1967IntegrationNorth-HollandAmsterdamGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • P. G. Dodds
    • 1
    Email author
  • T. K. Dodds
    • 1
  • F. A. Sukochev
    • 1
  • O. Ye. Tikhonov
    • 2
  1. 1.School of Informatics and EngineeringThe Flinders University of South AustraliaAdelaideAustralia
  2. 2.Research Institute of Mathematics and MechanicsKazan State UniversityKazanRussia

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