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 \(\ell ^1\).
Similar content being viewed by others
Notes
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.
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.
The James’ space from [6], being quasi-reflexive, has the dual RNP.
References
Castillo, J.M.F., Gonzalez, M.: Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667. Springer, Berlin (1997)
Diestel, J., Uhl, J.J.: Vector Measures. AMS, Providence (1977)
Dunford, N., Schwartz, J.T.: Linear Operators, Part III: Spectral Operators. Wiley, New York (1971)
Hagler, J.: Some more Banach spaces which contain \(L^1\). Studia Math. 46, 35–42 (1973)
Hadwin, D., Orhon, M.: A noncommutative theory of Bade functionals. Glasgow Math. J. 33, 73–81 (1991)
James, R.C.: Bases and reflexivity of Banach spaces. Ann. Math. 52(3), 518–527 (1950)
James, R.C.: A separable somewhat reflexive Banach space with nonseparable dual. Bull. Am. Math. Soc. 80, 738–743 (1974)
Kaijser, S.: Some representation theorems for Banach lattices. Ark. Mat. 16(2), 179–193 (1978)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Kitover, A., Orhon, M.: Reflexivity of Banach C(K)-modules via the reflexivity of Banach lattices. Positivity 18(3), 475–488 (2014)
Kitover, A., Orhon, M.: Weak sequential completeness in Banach C(K)-modules of finite multiplicity. Positivity 21(2), 739–753 (2017)
Lotz, H.P.: Minimal and reflexive Banach lattices. Math. Ann. 209, 117–126 (1973)
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)
Lozanovsky, GYa.: Some topological conditions on Banach lattices and reflexivity conditions on them. Soviet Math. Dokl. 9, 1415–1418 (1968)
Lozanovskii, G.Ya.: Banach structures and bases. Funct. Anal. Appl. 294(1), 92 (1967)
Lozanovskii, G.Ya.: Isomorphic Banach lattices. Sibirsk. Mat. Zh. 10(1), 93–98 (1969)
Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)
Orhon, M.: The ideal center of the dual of a Banach lattice. Positivity 14(4), 841–847 (2010)
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)
Rall, C.: Über Boolesche Algebren von Projektionen. Math Z. 153, 199–217 (1977)
Rall, C.: Boolesche Algebren von Projectionen auf Banachräumen. Ph.D. Thesis, Universität Tübingen (1977)
Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)
Stegall, C.: The Radon–Nikodym property in conjugate Banach spaces. Trans. Am. Math. Soc. 206, 213–223 (1975)
Uhl, J.J.: A note on the Radon–Nikodym property for Banach spaces. Rev. Roum. Math. Pures Appl. 17, 113–115 (1972)
Veksler, A.I.: Cyclic Banach spaces and Banach lattices. Soviet Math. Dokl. 14, 1773–1779 (1973)
Wnuk, W.: Banach Lattices with Order Continuous Norms. PWN, Warszawa (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-017-0529-2