# Norm-attaining lattice homomorphisms

### Sheldon Dantas

Universitat Jaume I, Castelló, Spain### Gonzalo Martínez-Cervantes

Universidad de Murcia, Spain### José David Rodríguez Abellán

Universidad de Murcia, Spain### Abraham Rueda Zoca

Universidad de Murcia, Spain

## Abstract

In this paper we study the structure of the set ${\rm Hom}(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice ${\rm FBL}(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$- and $C(K)$-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 $X$ and $C(K,X)$ attains its norm whenever $X$ 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. 981–1002

DOI 10.4171/RMI/1292