In this paper we study the structure of the set of all lattice homomorphisms from a Banach lattice into . Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice generated by a set contains a disjoint family of cardinality , answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as , - and -spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on and attains its norm whenever has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.
Cite this article
Sheldon Dantas, Gonzalo Martínez-Cervantes, José David Rodríguez Abellán, Abraham Rueda Zoca, Norm-attaining lattice homomorphisms. Rev. Mat. Iberoam. 38 (2022), no. 3, pp. 981–1002DOI 10.4171/RMI/1292