Skip to main content
Log in

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

  • Published:
Positivity Aims and scope Submit manuscript

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 \(\ell ^1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. If X and Y are Banach spaces we say that X contains a copy of Y if there is a closed subspace of X linearly isomorphic to Y.

  2. If X and Y are Banach lattices we say that X contains a copy of Y as a sublattice if there is a closed sublattice of X lattice isomorphic to Y.

  3. The James’ space from [6], being quasi-reflexive, has the dual RNP.

References

  1. Castillo, J.M.F., Gonzalez, M.: Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  2. Diestel, J., Uhl, J.J.: Vector Measures. AMS, Providence (1977)

    Book  MATH  Google Scholar 

  3. Dunford, N., Schwartz, J.T.: Linear Operators, Part III: Spectral Operators. Wiley, New York (1971)

    MATH  Google Scholar 

  4. Hagler, J.: Some more Banach spaces which contain \(L^1\). Studia Math. 46, 35–42 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hadwin, D., Orhon, M.: A noncommutative theory of Bade functionals. Glasgow Math. J. 33, 73–81 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. James, R.C.: Bases and reflexivity of Banach spaces. Ann. Math. 52(3), 518–527 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  7. James, R.C.: A separable somewhat reflexive Banach space with nonseparable dual. Bull. Am. Math. Soc. 80, 738–743 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kaijser, S.: Some representation theorems for Banach lattices. Ark. Mat. 16(2), 179–193 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  10. Kitover, A., Orhon, M.: Reflexivity of Banach C(K)-modules via the reflexivity of Banach lattices. Positivity 18(3), 475–488 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kitover, A., Orhon, M.: Weak sequential completeness in Banach C(K)-modules of finite multiplicity. Positivity 21(2), 739–753 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lotz, H.P.: Minimal and reflexive Banach lattices. Math. Ann. 209, 117–126 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lozanovsky, GYa.: Some topological conditions on Banach lattices and reflexivity conditions on them. Soviet Math. Dokl. 9, 1415–1418 (1968)

    Google Scholar 

  15. Lozanovskii, G.Ya.: Banach structures and bases. Funct. Anal. Appl. 294(1), 92 (1967)

  16. Lozanovskii, G.Ya.: Isomorphic Banach lattices. Sibirsk. Mat. Zh. 10(1), 93–98 (1969)

  17. Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  18. Orhon, M.: The ideal center of the dual of a Banach lattice. Positivity 14(4), 841–847 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  20. Rall, C.: Über Boolesche Algebren von Projektionen. Math Z. 153, 199–217 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rall, C.: Boolesche Algebren von Projectionen auf Banachräumen. Ph.D. Thesis, Universität Tübingen (1977)

  22. Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  23. Stegall, C.: The Radon–Nikodym property in conjugate Banach spaces. Trans. Am. Math. Soc. 206, 213–223 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Uhl, J.J.: A note on the Radon–Nikodym property for Banach spaces. Rev. Roum. Math. Pures Appl. 17, 113–115 (1972)

    MathSciNet  MATH  Google Scholar 

  25. Veksler, A.I.: Cyclic Banach spaces and Banach lattices. Soviet Math. Dokl. 14, 1773–1779 (1973)

    MATH  Google Scholar 

  26. Wnuk, W.: Banach Lattices with Order Continuous Norms. PWN, Warszawa (1999)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arkady Kitover.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kitover, A., Orhon, M. The dual Radon–Nikodym property for finitely generated Banach C(K)-modules. Positivity 22, 587–596 (2018). https://doi.org/10.1007/s11117-017-0529-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-017-0529-2

Keywords

Mathematics Subject Classification

Navigation