Dedekind Complete and Order Continuous Banach C(K)-Modules

  • Arkady KitoverEmail author
  • Mehmet Orhon
Part of the Trends in Mathematics book series (TM)


We extend the notions of Dedekind complete and σ-Dedekind complete Banach lattices to Banach C(K)-modules. As our main result we prove for these modules an analogue of Lozanovsky’s well-known characterization of Banach lattices with order continuous norm.


Banach lattice Dedekind complete Order continuous Banach C(K)-module 


  1. 1.
    Yu.A. Abramovich, E.L. Arenson, A.K. Kitover, in Banach C(K)-Modules and Operators Preserving Disjointness. Pitman Research Notes in Mathematical Series, vol. 277 (Longman Scientific and Technical, 1992)Google Scholar
  2. 2.
    J.M.F. Castillo, M. Gonzalez, in Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667 (Springer, 1997)Google Scholar
  3. 3.
    N. Dunford, J.T. Schwartz, Linear Operators, Part III: Spectral Operators (Wiley, New York, 1971)zbMATHGoogle Scholar
  4. 4.
    D. Hadwin, M. Orhon, Reflexivity and approximate reflexivity for Boolean algebras of projections. J. Funct. Anal. 87, 348–358 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D. Hadwin, M. Orhon, A noncommutative theory of Bade functionals. Glasgow Math. J. 33, 73–81 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    I. Kaplansky, Modules over operator algebras. Am. J. Math. 75(4), 839–853 (1953)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A.K. Kitover, M. Orhon, Reflexivity of Banach C(K)-modules via reflexivity of Banach lattices. Positivity 18(3), 475–488 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A.K. Kitover, M. Orhon, Weak sequential completeness in Banach C(K)-modules of finite multiplicity. Positivity 21(2), 739–753 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A.K. Kitover, M. Orhon, The dual Radon-Nikodym property for finitely generated Banach C(K)-modules. Positivity 22(2), 587–596 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G.Ya. Lozanovskii, On isomorphous Banach structures. Sib. Math. J. 10(1), 64–68 (1969)Google Scholar
  11. 11.
    H. Önder, M. Orhon, Transitive operator algebras on the n-fold direct sum of a cyclic Banach space. J. Oper. Theory 22, 99–107 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    C. Rall, Uber Boolesche Algebren von Projektionen. Math. Z. 153(3), 199–217 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    H.H. Schaefer, Banach Lattices and Positive Operators (Springer, Berlin, 1974)CrossRefGoogle Scholar
  14. 14.
    A.I. Veksler, Cyclic Banach spaces and Banach lattices. Sov. Math. Dokl. 14, 1773–1779 (1973)zbMATHGoogle Scholar
  15. 15.
    W. Wnuk, Banach Lattices with Order Continuous Norm (PWN, Warszawa, 1999)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations