Weak convergence in the dual of weak L^p

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

• (i) each T′, T ∈ $\mathcal{G}$, is a lattice homomorphism

• (ii) $\cup _{T \in \mathcal{G}}$[−u, u] contains the unit ball of E

• (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).