Positivity

, Volume 22, Issue 2, pp 587–596 | Cite as

The dual Radon–Nikodym property for finitely generated Banach C(K)-modules

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Abstract

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual \(X^\star \) has the RNP iff X does not contain a closed subspace isomorphic to \(\ell ^1\).

Keywords

Radon–Nikodym property Banach modules Banach lattices 

Mathematics Subject Classification

Primary 46B20 Secondary 47B22 46B42 

References

  1. 1.
    Castillo, J.M.F., Gonzalez, M.: Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  2. 2.
    Diestel, J., Uhl, J.J.: Vector Measures. AMS, Providence (1977)CrossRefMATHGoogle Scholar
  3. 3.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part III: Spectral Operators. Wiley, New York (1971)MATHGoogle Scholar
  4. 4.
    Hagler, J.: Some more Banach spaces which contain \(L^1\). Studia Math. 46, 35–42 (1973)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hadwin, D., Orhon, M.: A noncommutative theory of Bade functionals. Glasgow Math. J. 33, 73–81 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    James, R.C.: Bases and reflexivity of Banach spaces. Ann. Math. 52(3), 518–527 (1950)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    James, R.C.: A separable somewhat reflexive Banach space with nonseparable dual. Bull. Am. Math. Soc. 80, 738–743 (1974)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kaijser, S.: Some representation theorems for Banach lattices. Ark. Mat. 16(2), 179–193 (1978)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)MATHGoogle Scholar
  10. 10.
    Kitover, A., Orhon, M.: Reflexivity of Banach C(K)-modules via the reflexivity of Banach lattices. Positivity 18(3), 475–488 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kitover, A., Orhon, M.: Weak sequential completeness in Banach C(K)-modules of finite multiplicity. Positivity 21(2), 739–753 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lotz, H.P.: Minimal and reflexive Banach lattices. Math. Ann. 209, 117–126 (1973)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lotz, H.P., Rosenthal, H.P.: Embeddings of \(C(\Delta )\) and \(L^1(0,1)\) in Banach lattices. Israel J. Math. 31, 169–179 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lozanovsky, GYa.: Some topological conditions on Banach lattices and reflexivity conditions on them. Soviet Math. Dokl. 9, 1415–1418 (1968)Google Scholar
  15. 15.
    Lozanovskii, G.Ya.: Banach structures and bases. Funct. Anal. Appl. 294(1), 92 (1967)Google Scholar
  16. 16.
    Lozanovskii, G.Ya.: Isomorphic Banach lattices. Sibirsk. Mat. Zh. 10(1), 93–98 (1969)Google Scholar
  17. 17.
    Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  18. 18.
    Orhon, M.: The ideal center of the dual of a Banach lattice. Positivity 14(4), 841–847 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Orhon, M.: Algebras of operators containing a Boolean algebra of projections of finite multiplicity. In: Operators in Indefinite Metric Spaces, Scattering Theory And Other Topics (Bucharest, 1985). Oper. Theory: Adv. Appl., vol. 24, pp. 265–281. Birkhäuser, Basel (1987)Google Scholar
  20. 20.
    Rall, C.: Über Boolesche Algebren von Projektionen. Math Z. 153, 199–217 (1977)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Rall, C.: Boolesche Algebren von Projectionen auf Banachräumen. Ph.D. Thesis, Universität Tübingen (1977)Google Scholar
  22. 22.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefMATHGoogle Scholar
  23. 23.
    Stegall, C.: The Radon–Nikodym property in conjugate Banach spaces. Trans. Am. Math. Soc. 206, 213–223 (1975)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Uhl, J.J.: A note on the Radon–Nikodym property for Banach spaces. Rev. Roum. Math. Pures Appl. 17, 113–115 (1972)MathSciNetMATHGoogle Scholar
  25. 25.
    Veksler, A.I.: Cyclic Banach spaces and Banach lattices. Soviet Math. Dokl. 14, 1773–1779 (1973)MATHGoogle Scholar
  26. 26.
    Wnuk, W.: Banach Lattices with Order Continuous Norms. PWN, Warszawa (1999)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsCommunity College of PhiladelphiaPhiladelphiaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of New HampshireDurhamUSA

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