Weakly compactly generated Banach lattices

Abstract

Abstract. We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice X, the existence of a weakly compact set K ⊂ X such that X coincides with the band generated by K, implies that X is WCG. The general problemThe purpose of this note is to study Banach lattices which are generated in one way or another by a weakly compact set. Namely, we will explore the connection between the existence of a weakly compact set which generates a Banach lattice as a linear space, a lattice, an ideal or a band. Our motivation starts with the question of J. Diestel of whether every Banach lattice which is generated, as a lattice, by a weakly compact set must be weakly compactly generated (i.e., as a linear space).Recall that a Banach lattice is a Banach space endowed with additional order and lattice structures which behave well with respect to the norm and linear structure. This is in particular highlighted by the fact that x ≤ y whenever |x| ≤ |y|, or by the norm continuity of the lattice operations ∧ and ∨. However, for the weak topology, the relation with the order and lattice structures is more subtle, in particular it is not always true that the lattice operations are weakly continuous. In fact, on infinite dimensional Banach lattices the weak topology fails to be locally solid (see e.g. [1, Theorem 6.9]).A Banach space X is called weakly compactly generated (WCG) whenever there exists a weakly compact subset of X whose closed linear span coincides with X. This class of Banach spaces was first studied by Corson [12] and it was pushed further by the fundamental work of Amir and Lindenstrauss [5]. Nowadays, WCG spaces play a relevant role in non-separable Banach space theory. For complete information on WCG spaces, see [17,22,35].Weakly compact sets and weakly compact operators in Banach lattices have been the object of research by several authors (cf. [2,3,11,28]

Open problems in Banach spaces and measure theory

“…In the particular case of a Banach lattice X, Diestel [36] asked whether the property of being WCG is equivalent to the existence of a weakly compact set L ⊆ X such that the sublattice generated by L is dense in X. In [6] we answered in the affirmative this question for order continuous Banach lattices, but the general case still remains open.…”

The free Banach lattices generated by $$\ell _p$$ ℓ p and $$c_0$$ c 0

“…Moreover, a direct application of (1.1) yields that, for any set Γ, if p ∈ [1, 2], then F BL[ℓ p (Γ)] contains a subspace isomorphic to ℓ 1 (Γ), and thus, when Γ is uncountable, this space is not weakly compactly generated (in fact, it does not even embed as a subspace of a weakly compactly generated space). This provides the first known example of a Banach lattice which is weakly compactly generated as a Banach lattice but not as a Banach space, answering in the negative a question of J. Diestel (we refer to [3,4] for more details on this question).…”

“…. For(3), if the basis is symmetric, for each permutation σ of the natural numbers we have an isomorphism T σ : E −→ E such that T e n = e σ(n) . Extending this operator to a lattice homomorphism in F BL[E] we get statement (3).…”

The free Banach lattice generated by a Banach space

“…Is every LWCG Banach lattice WCG? This and related questions have been recently investigated in [3], where Problem 1.1 is solved affirmatively for Banach lattices which are order continuous or have weakly sequentially continuous lattice operations. Here we will provide a negative answer to Diestel's question by showing that the free Banach lattice F BL[ℓ 2 (Γ)] is LWCG but not WCG as long as Γ is uncountable (Corollary 5.5).…”