Extension of positive functionals on certain ordered spaces
type: Item
instance of: scholarly article
title: Extension of positive functionals on certain ordered spaces
zbMATH Open document ID: 0551.46005
DOI: 10.1007/BF02454158
author: C. T. Tucker
published in: Periodica Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: An ordered linear space L is said to satisfy extension property (E1) if for every directed subspace M of L and positive linear functional \(\Phi\) on M, \(\Phi\) can be extended to L. A Riesz space L is said to satisfy extension property (E2) if for every sub-Riesz space M of L and every real valued Riesz homomorphism \(\Phi\) on M, \(\Phi\) can be extended to L as a Riesz homomorphism. These properties were introduced by \textit{G. C. Schmidt}, Periodica Math. Hung. 6, 295-307 (1975; Zbl 0291.06010)]. 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 space L' defined on a set X' such that if f and g belong to L' there exists a finite disjoint subset M of the set of functions on X' such that each of f and g is a linear combination of the points of M. An analogous theorem is derived for Riesz spaces with extension property (E2).
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 3877777
zbMATH Keywords: ordered linear space
zbMATH Keywords: extension property
zbMATH Keywords: Riesz space
zbMATH Keywords: Riesz homomorphism
MaRDI profile type: Publication
cites work: Unique Representation of Archimedean Lattice Groups and Normal Archimedean Lattice Rings
cites work: Q5555984
cites work: Extension theorems for linear lattices with positive algebraic basis
description: scientific article; zbMATH DE number 3877777