Some remarks on Banach lattices with non-atomic duals.

*(English)*Zbl 0731.46008It 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)].

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)].

Reviewer: Yu.A.Abramovich (Indianapolis)

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Zbl 0439.46013
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\textit{B. de Pagter} and \textit{W. Wnuk}, Indag. Math., New Ser. 1, No. 3, 391--395 (1990; Zbl 0731.46008)

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##### 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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