On some properties of bilinear maps of order bounded variation
type: Item
instance of: scholarly article
title: On some properties of bilinear maps of order bounded variation
zbMATH Open document ID: 1113.47025
author: Karim Boulabiar
author: Robert Page
author: Gerard Buskes
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: The authors study properties of bilinear maps of order bounded variation and prove theorems about the preservation of properties when passing to the triadjoint and the tensor product. In particular, generalizations of some results of \textit{S.\,J.\,Bernau} and \textit{C.\,B.\,Huijsmans} [Trans.\ Am.\ Math.\ Soc.\ 347, No.\,11, 4259--4274 (1995; Zbl 0851.46003)], \textit{E.\,Scheffold} [Rev.\ Roum.\ Math.\ Pures Appl.\ 41, No.\,9--10, 697--701 (1996; Zbl 1112.46305); Czech.\ Math.\ J.\ 48, No.\,4, 747--753 (1998; Zbl 0954.46012)], and \textit{J.\,J.\,Grobler} [Positivity 3, No.\,4, 357--364 (1999; Zbl 0945.46003)] are obtained. One of the main tools in the proofs is an up-down-up-down theorem from Grobler's paper based upon a previous result of \textit{D.\,H.\,Fremlin} [Proc.\ Camb.\ Philos.\ Soc.\ 63, 653--660 (1967; Zbl 0149.33501)]. Unfortunately, insufficient care was given to editing the paper. Misprints and slips abound throughout Section 2, starting with the definition at the very beginning and ending with quotations (ii) and (iii) from the literature. The modulus of a bilinear map of order bounded variation used in the proof of Theorem 2.1(ii) is nowhere defined. There is a lapse concerning order continuity in the proof of Theorem 2.1(v). Besides, this proof seems too complicated. Finally, the assumption that \(F\) be Dedekind complete is missing in the formulation of Theorem 3.3.
reviewed by: Zbigniew Lipecki
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 5070768
zbMATH Keywords: Riesz space
zbMATH Keywords: Riesz homomorphism
zbMATH Keywords: bilinear map
zbMATH Keywords: Riesz biomorphism
zbMATH Keywords: Fremlin tensor product
zbMATH Keywords: Arens tri-adjoint
MaRDI profile type: Publication
full work available at URL: s11117-004-7544-9
OpenAlex ID: W2024440077
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DOI: 10.1007/S11117-004-7544-9
description: scientific article; zbMATH DE number 5070768