Representations of positive projections. I
type: Item
instance of: scholarly article
title: Representations of positive projections. I
zbMATH Open document ID: 1119.47038
author: B. de Pagter
author: W. A. J. Luxemburg
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: This paper develops a general theory of representations of positive projections on unital f-algebras and Dedekind complete vector lattices motivated by the work of \textit{D.\,Maharam} [Trans.\ Am.\ Math.\ Soc.\ 65, 279--330 (1949; Zbl 0036.31401)]. Preliminary information is provided as well as a discussion of order separability and a weaker notion of locally order separable. A measurable space \((X,\Sigma)\) is called a representation space for a Dedekind \(\sigma\)-complete Riesz space \(L\) with weak order unit \(w\) if there is an ideal \(\widehat L\) in the measurable functions and containing \(1\) and a \(\sigma\)-order continuous Riesz homomorphism \(\Phi\) from \(\widehat L\) onto \(L\) where the image of \(1\) is the weak order unit. It is shown that there indeed exists a representation space for \(L \). Now a Maharam-type representation space for a positive projection \(P\) on \(L\) to a Riesz subspace of \(L\) containing \(w\) is considered. In particular, if a product of two measurable spaces \((X,\Sigma)\) and \((Y,\Lambda)\) such that the product \((X \times Y)\) with the natural measurable sets is a representation space for \(L\) (via a map \(\Phi\)) and there is an appropriate scalar measure \(\mu\) on \(Y\) and measurable function \(m\) on the product with \(Rf(x,y)=\int_Y m(x,z)f(x,z)\,d\mu(z)\) representing \(P\) in the sense that \(P(\Phi f)=\Phi(Rf)\), then this product space is called a Maharam-type representation space. For the construction of representation spaces for projections, it is established that one need only consider spaces with strong order units. Since these spaces have the structure of \(f\)-algebras, while complete Riesz subspaces containing the unit are \(f\)-subalgebras, the authors analyze projections in this setting. For an appropriate projection onto an \(f\)-subalgebra \(B\) containing \(1\), if there exists a complete \(f\)-subalgebra \(C\) containing \(B\), then there exists a unique positive projection \(Q\) onto \(C\) …
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 5070764
reviewed by: William A. Feldman
MaRDI profile type: Publication
full work available at URL: s11117-004-2773-5
OpenAlex ID: W4248306561
cites work: Subalgebras and Riesz Subspaces of an f -Algebra
cites work: Averaging operators and positive contractive projections
cites work: Q3972267
cites work: An Extension of the Concept of the Order Dual of a Riesz Space
cites work: Maharam extensions of positive operators and \(f\)-modules
cites work: Representations of positive projections. II
DOI: 10.1007/S11117-004-2773-5
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description: scientific article; zbMATH DE number 5070764