Some remarks on Banach lattices with non-atomic duals
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)].
Dedekind \(\sigma \) -complete Banach lattice
scientific article; zbMATH DE number 4208821
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