Norm-attaining lattice homomorphisms

TL;DR

This paper investigates the structure and norm-attainment properties of lattice homomorphisms from Banach lattices to real numbers, revealing new insights into their dual spaces, norm-attaining behavior, and limitations of approximation theorems.

Contribution

It characterizes the structure of lattice homomorphisms, constructs examples of non-norm-attaining homomorphisms, and shows the failure of Bishop-Phelps type theorems in this setting.

Findings

The dual of free Banach lattices contains large disjoint families.

Classical Banach lattices have all lattice homomorphisms attain their norm.

Counterexamples exist of lattice homomorphisms that do not attain their norm.

Abstract

In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ 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 $FBL(A)$ 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.