Abstract
Properties of Schur type for Banach lattices of regular operators and tensor products are analyzed. It is shown that the dual positive Schur property behaves well with respect to Fremlin’s projective tensor product, which allows us to construct new examples of spaces with this property. Similar results concerning the positive Grothendieck property are also presented.
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References
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, New York (2006)
Aqzzouz, B., Elbour, A., Wickstead, A.W.: Positive almost Dunford–Pettis operators and their duality. Positivity 15, 185–197 (2011)
Bu, Q., Buskes, G., Popov, A.I., Tcaciuc, A., Troitsky, V.G.: The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square. Positivity 17(2), 283–298 (2013)
Diestel, J.: A survey of results related to the Dunford–Pettis property. In: Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. 15–60, Contemp. Math., 2, Am. Math. Soc. (1980)
Flores, J., Hernández, F.L., Spinu, E., Tradacete, P., Troitsky, V.G.: Disjointly homogeneous Banach lattices: duality and complementation. J. Funct. Anal. 266, 5858–5885 (2014)
Fremlin, D.H.: Tensor products of Archimedean vector lattices. Am. J. Math. 94, 777–798 (1972)
Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)
Ji, D., Craddock, M., Bu, Q.: Reflexivity and the Grothendieck property for positive tensor products of Banach lattices. I. Positivity 14(1), 59–68 (2010)
Labuschagne, C.C.A.: Riesz reasonable cross norms on tensor products of Banach lattices. Quaest. Math. 27(3), 243–266 (2004)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. Springer, New York (1979)
Meyer-Nieberg, P.: Banach Lattices. Springer, New York (1991)
Troitsky, V.G., Zabeti, O.: Fremlin tensor products of concavifications of Banach lattices. Positivity 18(1), 191–200 (2014)
Wickstead, A.W.: AL-spaces and AM-spaces of operators. Positivity and its applications (Ankara, 1998). Positivity 4(3), 303–311 (2000)
Wnuk, W.: Some remarks on the positive Schur property in spaces of operators. Funct. Approx. Comment. Math. 21, 65–68 (1992)
Wnuk, W.: Banach lattices with properties of the Schur type—a survey. Conferenze del Seminario di Matematica dell’Università di Bari 249, 1–24 (1993)
Wnuk, W.: On the dual positive Schur property in Banach lattices. Positivity 17(3), 759–773 (2013)
Acknowledgments
Most of the work on this paper was done during the author’s visit to the Adam Mickiewicz University. He wishes to thank the Department of Functional Analysis, and specially Professor W. Wnuk, for their great hospitality.
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This research has been supported by the Spanish Government through grants MTM2010-14946 and MTM2012-31286, as well as Grupo UCM 910346.
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Tradacete, P. Positive Schur properties in spaces of regular operators. Positivity 19, 305–316 (2015). https://doi.org/10.1007/s11117-014-0296-2
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DOI: https://doi.org/10.1007/s11117-014-0296-2
Keywords
- Banach lattice
- Positive Schur property
- Positive Grothendieck property
- Spaces of regular operators
- Fremlin tensor product