Israel Journal of Mathematics

, Volume 214, Issue 2, pp 867–884 | Cite as

On Markushevich bases in preduals of von Neumann algebras

  • Martin Bohata
  • Jan Hamhalter
  • Ondřej F. K. Kalenda


We prove that the predual of any von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U. Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the selfadjoint part of the predual is 1-Plichko as well.


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  1. [1]
    D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Annals of Mathematics 88 (1968), 35–46.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. Argyros, S. Mercourakis and S. Negrepontis, Functional-analytic properties of Corsoncompact spaces, Studia Mathematica 89 (1988), 197–229.MathSciNetzbMATHGoogle Scholar
  3. [3]
    M. Cúth, Simultaneous projectional skeletons, Journal of Mathematical Analysis and Applications 411 (2014), 19–29.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Deville and G. Godefroy, Some applications of projective resolutions of identity, Proceedings of the London Mathematical Society 67 (1993), 183–199.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Ghoussoub, G. Godefroy, B. Maurey and W. Schachermayer, Some topological and geometrical structures in Banach spaces, Memoirs of the American Mathematical Society 70 (1987).Google Scholar
  6. [6]
    R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Graduate Studies in Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1997.Google Scholar
  7. [7]
    O. F. K. Kalenda, Valdivia compact spaces in topology and Banach space theory, Extracta Mathematicae 15 (2000), 1–85.MathSciNetzbMATHGoogle Scholar
  8. [8]
    O. F. K. Kalenda, Note on Markushevich bases in subspaces and quotients of Banach spaces, Polish Academy of Sciences. Bulletin. Mathematics 50 (2002), 117–126.MathSciNetzbMATHGoogle Scholar
  9. [9]
    O. F. K. Kalenda, Complex Banach spaces with Valdivia dual unit ball, Extracta Mathematicae 20 (2005), 243–259.MathSciNetzbMATHGoogle Scholar
  10. [10]
    O. F. K. Kalenda, Natural examples of Valdivia compact spaces, Journal of Mathematical Analysis and Applications 340 (2008), 81–101.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    W. Kubiś, Banach spaces with projectional skeletons, Journal of Mathematical Analysis and Applications 350 (2009), 758–776.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. Lindenstrauss, On reflexive spaces having the metric approximation property, Israel Journal of Mathematics 3 (1965), 199–204.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Lindenstrauss, On nonseparable reflexive Banach spaces, Bulletin of the American Mathematical Society 72 (1966), 967–970.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Mercourakis, On weakly countably determined Banach spaces, Transactions of the American Mathematical Society 300 (1987), 307–327.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. N. Plichko, E-mail to the third author, December 1999.Google Scholar
  16. [16]
    A. N. Plichko, Projection decompositions of the identity operator and Markushevich bases, Doklady Akademii Nauk SSSR 263 (1982), 543–546.MathSciNetGoogle Scholar
  17. [17]
    S. Sakai, C*-Algebras and W*-Algebras, Classics in Mathematics, Springer-Verlag, Berlin–Heidelberg–New York, 1998.Google Scholar
  18. [18]
    M. Takesaki, Theory of Operator Algebras. I, Springer-Verlag, New York–Heidelberg, 1979.CrossRefzbMATHGoogle Scholar
  19. [19]
    M. Valdivia, Projective resolution of identity in C (K) spaces, Archiv der Mathematik 54 (1990), 493–498.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collectanea Mathematica 42 (1991), 265–284 (1992).MathSciNetzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  • Martin Bohata
    • 1
  • Jan Hamhalter
    • 1
  • Ondřej F. K. Kalenda
    • 2
  1. 1.Czech Technical University in PragueFaculty of Electrical Engineering, Department of MathematicsPrague 6Czech Republic
  2. 2.Charles University in PragueFaculty of Mathematics and Physics, Department of Mathematical AnalysisPraha 8Czech Republic

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