On some properties of bilinear maps of order bounded variation
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.
scientific article; zbMATH DE number 5070768