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Preregular maps between Banach lattices

Published online by Cambridge University Press:  17 April 2009

David A. Birnbaum
David A. Birnbaum
Affiliation:
Department of Mathematics, Amherst College, Amherst, Massachusetts, USA.
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Abstract

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A continuous linear map from a Banach lattice E into a Banach lattice F is preregular if it is the difference of positive continuous linear maps from E into the bidual F″ of F. This paper characterizes Banach lattices B with either of the following properties:

(1) for any Banach lattice E, each map in L(E, B) is preregular;

(2) for any Banach lattice F, each map in L(B, F) is preregular.

It is shown that B satisfies (1) (repectively (2)) if and only if B′ satisfies (2) (respectively (1)). Several order properties of a Banach lattice satisfying (2) are discussed and it is shown that if B satisfies (2) and if B is also an atomic vector lattice then B is isomorphic as a Banach lattice to 11(Γ) for some index set Γ.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 2 , October 1974 , pp. 231 - 254
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Andô, T., “On fundamental properties of a Banach space with a cone”, Pacific J. Math. 12 (1962), 1163–1169.CrossRefGoogle Scholar
[2]Crenshaw, J.A., “Extreme positive linear operators”, Math. Scand. 25 (1969), 195–217.CrossRefGoogle Scholar
[3]Grothendieck, Alexandra, Produits tensoriels topologiques et espaces nucléaires (Mem. Amer. Math. Soc. 16. Amer. Math. Soc., Providence, Rhode Island, 1955).CrossRefGoogle Scholar
[4]Jacobs, Harold, “Ordered topological tensor products”, Dissertation, University of Illinois, Urbana, Illinois, 1969.Google Scholar
[5]Lindenstrauss, J. and Petczyński, A., “Absolutely summing operators in Lp spaces and their applications”, Studia Math. 29 (1968), 275–326.CrossRefGoogle Scholar
[6]Lotz, H.P., Lectures on topological tensor products, linear mappings and nuclear spaces (University of Illinois, Urbana, Illinois, 1971).Google Scholar
[7]Marti, Jürg T., Introduction to the theory of bases (Springer Tracts in Natural Philosophy, 18. Springer-Verlag, Berlin, Heidelberg, New York, 1969).CrossRefGoogle Scholar
[8]Namioka, Isaac, Partially ordered linear topological vector spaces (Mem. Amer. Math. Soc. 24. Amer. Math. Soc., Providence, Rhode Island, 1957).Google Scholar
[9]Pełczyński, A., “Projections in certain Banach spaces”, Studia Math. 19 (1960), 209–228.CrossRefGoogle Scholar
[10]Peressini, Anthony L., Ordered topological vector spaces (Harper and Row, New York, Evanston, and London, 1967).Google Scholar
[11]Peressini, A.L. and Sherbert, D.R., “Ordered topological tensor products”, Proc. London Math. Soc. (3) 19 (1969), 177–190.CrossRefGoogle Scholar
[12]Pietsch, Albrecht, Nukleare lokalkonvexe Räume (Akademie-Verlag, Berlin, 1965; 2nd ed., Akademie-Verlag, Berlin, 1969). English translation of 2nd ed.: Nuclear locally convex spaces (translated by Ruckle, William H.. Ergebnisse der Mathematik und ihrer Grenzgebiete, 66. Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[13]Pietsch, A., “Absolut p–summierende Abbildungen in normierten Räumen”, Studia Math. 28 (1967), 333–353.CrossRefGoogle Scholar
[14]Pietsch, Albrecht, “Zur Theorie der topologischen Tensorprodukte”, Math. Nachr. 25 (1963), 19–30.CrossRefGoogle Scholar
[15]Schaefer, Helmut H., Topological vector spaces (The Macmillan Company, New York; Collier-Macmillan, London, 1966).Google Scholar
[16]Schlotterbeck, Ulf, “Über Klassen majorisierbarer Operatoren auf Banachverbänden”, Dissertation, Eberhard-Karls-Universität zu Tübingen, Tübingen, 1969.Google Scholar