Ideals and bands in principal modules
type: Item
instance of: scholarly article
title: Ideals and bands in principal modules
zbMATH Open document ID: 0581.47024
DOI: 10.1007/BF01198233
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: The main purpose of the paper is to present abstract versions in the framework of the theory of principal modules of some results of \textit{B. de Pagter} [Indag. Math. 45, 229-241 (1983; Zbl 0521.47018)] and \textit{C. D. Aliprantis} and \textit{O. Burkinshaw} [Math. Z. 184, 245-257 (1983; Zbl 0503.47035)]. For the reader's convenience, the paper begins with a self-contained introduction to the author's theory of principal modules; thus, the section reproduces some main results of the theory published by the author in some earlier papers. Recall that a principal module is a topological Riesz space E which is a module over an f-algebra A with a strong order unit such that the submodule generated by an arbitrary \(x\in E\) is dense in the principal order ideal generated by x. Among the results in the section we mention: a) The characterization of principal modules: an Archimedean Riesz space E endowed with a locally solid topology and with a structure of A-module is principal iff for every \(x_ 1,x_ 2\in E\) with \(x_ 1\wedge x_ 2=0\) and every neighborhood W of O and E there are \(a_ 1,a_ 2\in A\) with \(a_ 1\wedge a_ 2=0\) and \(x_ i-a_ ix_ i\in W\) \((i=1,2);\) b) The permanence theorem (proved in a preprint from 1981 and stated (without proof) in some papers from 1983-1984): let E be a principal A- module and let F be a principal B-module whose topology is order continuous. Then \(L'_ r(E,F)\) is a principal A\({\bar \otimes}B\)-module for the solid strong topology. The next section presents monotone approximation theorems in principal modules. For instance, it is proved that in a principal module E whose topology is separated and Fatou, we have \([-| x|,| x|]=([-e,e]x)^{0\omega \omega}\) for any \(x\in E\) (e denotes the unit of A and \(M^ 0\) (respectively \(M^{\omega})\) stands for the set of limits of order convergent sequences (respectively nets) in M). As a particular case, de Pagter's result on the monotone approximation of the components of a positive ope…
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 3929766
zbMATH Keywords: regular operator
zbMATH Keywords: principal modules
zbMATH Keywords: topological Riesz space
zbMATH Keywords: f-algebra
zbMATH Keywords: principal order ideal
zbMATH Keywords: locally solid topology
zbMATH Keywords: permanence theorem
zbMATH Keywords: solid strong topology
zbMATH Keywords: monotone approximation theorems
zbMATH Keywords: limits of order convergent sequences
zbMATH Keywords: band
zbMATH Keywords: Radon-Nikodym theorem
author: Dan Tudor Vuza
MaRDI profile type: Publication
cites work: Groupes et anneaux reticules
cites work: Tensor Products of Archimedean Vector Lattices
cites work: Subalgebras and Riesz Subspaces of an f -Algebra
cites work: Q3671733
cites work: Q3772783
cites work: Q3763072
cites work: Q3662120
cites work: Q3343180
cites work: Q3712787
full work available at URL: https://doi.org/10.1007/bf01198233
OpenAlex ID: W2016214120
description: scientific article; zbMATH DE number 3929766