The free Banach lattice generated by a Banach space

Abstract

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.

The free Banach lattice generated by a lattice

“…(1) L is distributive, (2) L is lattice-isomorphic to a subset of a Boolean algebra, (3) L is lattice-isomorphic to a bounded subset of a Banach lattice, (4) The canonical map φ : L −→ F BL L is injective.…”

“…Proof. The equivalence of (1), (2) and (3) is well known, see [4,Theorem II.19] for 1 ⇒ 2, [5, Theorem 1.b.3] for 2 ⇒ 3 and [7, Proposition II.1.5] for 3 ⇒ 1. It is obvious that (4) implies (3).…”

“…The free Banach lattice generated by a set A with no extra structure, which is denoted by F BL(A), has been recently introduced and analyzed by B. de Pagter and A.W. Wickstead in [6], while the free Banach lattice generated by a Banach space E has been studied by A. Avilés, J. Rodríguez and P. Tradacete in [2]. We can consider a similar idea using lattices instead of Banach spaces.…”

On projective Banach lattices of the form C(K) and FBL[E]

“…The notion of free Banach lattice was also introduced in [9]. If A is a set with no extra structure, the free Banach lattice generated by A, denoted by F BL(A), is a Banach lattice together with a bounded map u : A −→ F BL(A) having the following universal In [3] and [9] it is shown that both objects exist and are unique up to Banach lattices isometries. A more explicit description of these spaces is given in [3] as follows:…”

“…From this theorem we can deduce for example that no Banach lattice ℓ p is λ-projective, for any λ > 0 or p = 1. The prototype of 1 + -projective Banach lattice is F BL(A) = F BL[ℓ 1 (A)] (see [3,Corollary 2.8] and [9, Proposition 10.2]), so it would be natural to wonder whether F BL[E] might be 1 + -projective as well for other Banach spaces E. We show that, for this to happen, the structure of E must be very close to that of ℓ 1 (A):…”