Norm-attaining lattice homomorphisms and renormings of Banach lattices

TL;DR

This paper investigates norm-attaining lattice homomorphisms in Banach lattices, revealing that such properties are not preserved under isomorphisms and exploring renormings that control which homomorphisms attain their norm.

Contribution

It characterizes when lattice homomorphisms attain their norm and constructs renormings to isolate specific norm-attaining homomorphisms, addressing open questions.

Findings

In AM-spaces, all lattice homomorphisms attain their norm.

Existence of equivalent norms on $C(K)$ spaces where some homomorphisms do not attain their norm.

Construction of renormings where only coordinate functionals attain their norm.

Abstract

A well-known theorem due to R. C. James states that a Banach space is reflexive if and only if every bounded linear functional attains its norm. In this note we study Banach lattices on which every (real-valued) lattice homomorphism attains its norm. Contrary to what happens in the Banach space setting, we show that this property is not invariant under lattice isomorphisms. Namely, we show that in an AM-space every lattice homomorphism attains its norm, whereas every infinite-dimensional $C(K)$ space admits an equivalent lattice norm with a lattice homomorphism which does not attain its norm. Furthermore, we characterize coordinate functionals of atoms and show that whenever a Banach lattice $X$ supports a strictly positive functional, there exists a renorming with the property that the only (non-trivial) lattice homomorphisms attaining their norm are precisely these coordinate functionals. As a consequence, one can exhibit examples of Dedekind complete Banach lattices admitting a renorming with a non-norm-attaining lattice homomorphism, answering negatively questions posed by Dantas, Rodr\'iguez Abell\'an, Rueda Zoca and the fourth author.