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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

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