LATTICE EMBEDDINGS IN FREE BANACH LATTICES OVER LATTICES

Any lattice embedding L -> M between two lattices L subset of M induces a Banach lattice homomorphism (i) over cap: FBL < L > -> FBL < M > between the corresponding free Banach lattices generated by these lattices. We show that this mapping (i) over cap might not be an isomorphic embedding. Sufficient conditions for (i) over cap to be an isometric embedding are provided by considering a notion of locally complemented lattices. As a consequence, we obtain that every free Banach lattice generated by a lattice is Banach lattice isomorphic to an AM-space. Furthermore, we prove that (i) over cap 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] ex tends to a lattice homomorphism (x) over cap*: M -> [-1, 1]. Using this characterization we provide an example of lattices L subset of M for which I is an isomorphic not isometric embedding.