## Some remarks on Banach lattices with non-atomic duals.(English)Zbl 0731.46008

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)].

### MSC:

 46B42 Banach lattices 46A40 Ordered topological linear spaces, vector lattices

Zbl 0439.46013
Full Text:

### References:

 [1] Jonge, E.de, The semi-M-property for normed Riesz spaces, Compositio math., 34, 147-172, (1977) · Zbl 0345.46009 [2] Lozanovskii, G.Ja., Discrete functionals in Marcinkiewicz and Orlicz spaces (Russian), (), 132-147 [3] Luxemburg, W.A.J.; Zaanen, A.C., Riesz spaces I, (1971), North-Holland Amsterdam-London · Zbl 0231.46014 [4] Schaefer, H.H., Banach lattices and positive operators, (1974), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0296.47023 [5] Zaanen, A.C., Riesz spaces II, (1983), North-Holland Amsterdam-New York-Oxford · Zbl 0519.46001
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