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On the Banach spaces with the property \((V^*)\) of Pelczynski. II. (English) Zbl 0759.46018

A bounded subset \(X\) of a Banach space \(E\) is called a \(V^*\) set if \(\lim_n \sup_x | x_n^*(x)| = 0\) for every sequence \((x_n^*)\) in \(E^*\) for which the series \(\sum x_n^*\) is weakly unconditionally convergent. The Banach space \(E\) is said to have Property \(V^*\) [A. Pełczyński, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 10, 641–648 (1962; Zbl 0107.32504)] if every \(V^*\) set is relatively weakly compact. The author proves that \(E\) has Property \(V^*\) if and only if any conjugate unconditionally converging operator from \(E^*\) to \(F^*\) (\(F\) an arbitrary Banach space) is weakly compact. The sufficiency is proved here, the necessity was given in an earlier paper [part I, Ann. Mat. Pura Appl., IV. Ser. 152, 171–181 (1988; Zbl 0677.46006)]. This result leads to a long list of equivalent properties for a Banach space which is a complemented subspace of a Banach lattice. The list includes:
(i) \(E^*\) has Property \(V^*\),
(ii) \(E^*\) does not contain a copy of \(c_0\), and
(iii) \(E\) does not contain a complemented copy of \(\ell_1\).
The paper also explores other relationships among these and related properties.

MSC:

46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices
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References:

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