Lattice embeddings in free Banach lattices over lattices

Resumen

In this article we deal with the free Banach lattice generated by a lattice and its behavior with respect to subspaces. In general, any lattice embedding i: L −→ Mbetween two lattices L ⊆ M induces a Banach lattice homomorphism ˆı: F BLhLi −→ F BLhMi between the corresponding free Banach lattices. We show that this mappingˆı might not be an isometric embedding neither an isomorphic embedding. In order to provide sufficient conditions for ˆı to be an isometric embedding we define the notion oflocally complemented lattices and prove that, if L is locally complemented in M, then ˆıyields an isometric lattice embedding from F BLhLi into F BLhMi. We provide a widenumber of examples of locally complemented sublattices and, as an application, we obtain that every free Banach lattice generated by a lattice is lattice isomorphic to an AM-spaceor, equivalently, to a sublattice of a C(K)-space. Furthermore, we prove that ˆı is an isomorphic embedding if and only if it is injective, which in turn is equivalent to the fact that every lattice homomorphism x ∗ : L −→ [−1, 1] extends to a lattice homomorphism ˆx ∗ : M −→ [−1, 1]. Using this characterization weprovide an example of lattices L ⊆ M for which ˆı is an isomorphic not isometric embedding.