An alternative proof of a Radon-Nikodým theorem for lattice homomorphisms
type: Item
instance of: scholarly article
title: An alternative proof of a Radon-Nikodým theorem for lattice homomorphisms
zbMATH Open document ID: 0807.47023
DOI: 10.1007/BF00046637
published in: Acta Applicandae Mathematicae
publication date: 1993-09-30T00:00:00Z
review text: Let \(E\) be an Archimedean and \(F\) a Dedekind complete vector lattice. Then the space \(L_ b(E,F)\) of order bounded linear operators from \(E\) to \(F\) is also a Dedekind complete lattice, but in general the evaluation mapping \(T\to Tu\) \((u>0)\) is not a lattice homomorphism. So the crucial result of this paper is of great interest: Let \(T\) be a lattice homomorphism from \(E\) to \(F\), and let \(0\leq U\), \(V\) be arbitrary elements of the band generated by \(T\) in \(L_ b(E,F)\). Then \((U\wedge V)(u)= U(u)\wedge V(u)\) for all \(u\geq 0\). As an application the authors get a new proof of the following result due to the second author and \textit{A. R. Schep} [Indag. Math. 40, 357-375 (1978; Zbl 0389.47018)]: Let again \(T\) be a lattice homomorphism from \(E\) to \(F\). Then the following assertions are equivalent: (i) \(S\in \{T\}^{dd}\). (ii) \(Su\in \{Tu\}^{dd}\) for all \(u\geq 0\). Here \(\{x\}^{dd}\) denotes the band generated by \(x\).
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47L05
zbMATH DE Number: 417084
zbMATH Keywords: Dedekind complete vector lattice
zbMATH Keywords: order bounded linear operators
zbMATH Keywords: lattice homomorphism
author: Charles B. Huijsmans
author: W. A. J. Luxemburg
reviewed by: Manfred P. H. Wolff
MaRDI profile type: Publication
cites work: Orthomorphisms of archimedean vector lattices
cites work: Q3977662
cites work: Q4171180
cites work: Q4100960
cites work: Q4162108
full work available at URL: https://doi.org/10.1007/bf00046637
OpenAlex ID: W4243146157
description: scientific article; zbMATH DE number 417084