Some remarks on Banach lattices with non-atomic duals
type: Item
instance of: scholarly article
title: Some remarks on Banach lattices with non-atomic duals
zbMATH Open document ID: 0731.46008
DOI: 10.1016/0019-3577(90)90026-J
author: B. de Pagter
author: Witold Wnuk
published in: Indagationes Mathematicae. New Series
publication date: 1990-01-01T00:00:00Z
review text: It is a well known result due to G. Lozanovsky that a dual space \(E^*\) of a Dedekind \(\sigma\)-complete Banach lattice E is non-atomic (1) iff for each \(0\leq x\in E\) and each \(\epsilon >0\) there exist \(x_ 1,...,x_ n\in E_+\) such that \(x_ i\wedge x_ j=0\) (i\(\neq j)\), \(\| x_ i\| <\epsilon\) and \(\sum x_ i=x(2).\) If E is a Banach lattice, then still (2)\(\Rightarrow (1)\), but the validity of the converse implication is unknown. The authors find a nice generalization of this result for arbitrary Banach lattices by proving that \((1)\Leftrightarrow (2')\), where \((2')\) means that for each \(0\leq x\in E\) and each \(\epsilon >0\) there exist \(x_ 1,...,x_ n\in E_+\) such that \(\| x_ i\| <\epsilon\) and \(x\leq \bigvee^{n}_{i=1}x_ i.\) At the end of the paper this result is applied to obtain an unexpected Proposition 6, asserting that any semi-M-space (in the sense of de Jonge) with a non-atomic dual necessarily has a \(\sigma\)-order continuous norm. Reviewer's remarks. The functional p(x) which is essential for the proof of Theorem 1 was introduced by the reviewer in [Vestn. Leningrad.Univ. Math. 4, 153-159 (1977)]. Notice also that in the definition of semi-M- spaces it is possible to replace arbitrary elements by disjoint ones and this simplifies the investigation of this spaces considerably. This was done in another paper by the reviewer [Funkt. Anal., Ul'janovsk 13, 3-10 (1979; Zbl 0439.46013)].
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 4208821
zbMATH Keywords: dual space
zbMATH Keywords: Dedekind \(\sigma \) -complete Banach lattice
zbMATH Keywords: semi-M-space
reviewed by: Y. A. Abramovich
MaRDI profile type: Publication
cites work: Q4114322
cites work: Q3903514
cites work: Q5638775
cites work: Q4049667
cites work: Q3669960
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description: scientific article; zbMATH DE number 4208821