Extension of positive functionals on certain ordered spaces |
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Authors: | C T Tucker |
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Institution: | (1) Department of Mathematics, University of Houston, 77004 Houston, TX, USA |
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Abstract: | An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL.
A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism.
These properties were introduced by Schmidt in 5].
In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic
to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM.
An analogous theorem is derived for Riesz spaces with extension property (E2). |
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Keywords: | Primary 46A40 |
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