Representations of positive projections. II
type: Item
instance of: scholarly article
title: Representations of positive projections. II
zbMATH Open document ID: 1119.47039
author: B. de Pagter
author: W. A. J. Luxemburg
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: [Part I appeared ibid., No.\,3, 293--325 (2005; Zbl 1119.47038), see the preceding review.] The authors continue the study presented in Part I of this two-part series of papers. The present paper develops the authors' results to facilitate the creation of scalar-valued measures needed for the Maharam-type integral representations (defined in Part I) of positive projections on Dedekind complete Riesz spaces. A vector-valued version of a Radon--Nikodym type theorem due to \textit{D.\,Maharam} [Trans.\ Am.\ Math.\ Soc.\ 69, 142--160 (1950; Zbl 0041.18002 )] is established. It is shown that for a Dedekind complete \(f\)-algebra \(A\) with unit and \(P\) a positive order continuous projection onto a \(f\)-subalgebra \(B\) with unit, that if \(A\) is nowhere full with respect to \(B\) (discussed in Part I), \(e\) is a nonzero component of the unit in \(A\), and \(g\) is a positive element in \(B\) with \(g\leq P(e)\), then there exists a component \(p\) of the unit in \(A\) such that \(P(p)=g\). It follows that for any \(\lambda\) in \([0,1]\) there is a component \(p\) of the unit in \(A\) with \(P(p)=\lambda 1\). Properties of the set all such components and its complete Boolean subalgebras are studied. These lead to the creation of the appropriate measure spaces and measures. Indeed, given the same hypotheses as above with the additional property that \(A\) is order separable and a representation for \(B\), then there exists a representation space for the projection \(P\) and the algebra \(A\) compatible with the representation for \(B\). Consequences for Riesz spaces are presented, including assuming only a locally order separable condition and excluding the assumption of a weak order unit.
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 5070779
reviewed by: William A. Feldman
MaRDI profile type: Publication
full work available at URL: s11117-004-2774-4
OpenAlex ID: W2074547183
cites work: Q3748810
cites work: Representations of positive projections. I
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cites work: The Representation of Abstract Measure Functions
cites work: Decompositions of Measure Algebras and Spaces
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DOI: 10.1007/S11117-004-2774-4
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description: scientific article; zbMATH DE number 5070779