Abstract
We consider a Dedekind σ-complete Banach lattice E whose dual is weakly sequentially complete. Suppose that E has a positive element u and a family of positive operators \( \mathcal{G} \) such that
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(i)
each T′, T ∈ \( \mathcal{G} \), is a lattice homomorphism
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(ii)
\( \cup _{T \in \mathcal{G}} \)[−u, u] contains the unit ball of E
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(iii)
for any sequence (x n ) ⊂ [0, u] of pairwise disjoint elements and for any sequence (T n ) ⊂ \( \mathcal{G} \) the sequence (T n x n ) is majorized in E.
We show that such a space is a Grothendieck space, i.e., in the dual every weak* convergent sequence converges weakly (Theorem 1). We prove that Weak L p on a real interval satisfies the conditions above if 1 < p < ∞ (Theorem 2). Then we show that every Weak L p space is a Grothendieck space (Theorem 3).
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The author is indebted to the referee for his detailed comments and valuable suggestions
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Lotz, H.P. Weak convergence in the dual of weak Lp . Isr. J. Math. 176, 209–220 (2010). https://doi.org/10.1007/s11856-010-0026-9
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DOI: https://doi.org/10.1007/s11856-010-0026-9