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.
Highlights
It is well-known that in a Banach space E, the set of all continuous linear functionals from E into R determines almost totally the structure of E as a Banach space
Notice that positive linear functionals are functionals which respect the order in the Banach lattice, but they do not need to respect the lattice operations; in this paper we focus on the much more restrictive subclass of the set of positive linear functionals x∗ on X∗ for which, both equalities x∗(x ∨ y) = x∗(x) ∨ x∗(y) and x∗(x ∧ y) = x∗(x) ∧ x∗(y) hold for every x, y ∈ X
We characterize lattice homomorphisms attaining their norm on F BL[E] whenever E is an isometric predual of 1(A) or is isometric to 1(A) for some infinite set A. These results allow us to show that no Bishop-Phelps theorem holds in the class of Banach lattices, i.e. that there are lattice homomorphisms which cannot be approximated in norm by norm-attaining lattice homomorphisms
Summary
It is well-known that in a Banach space E, the set of all continuous linear functionals from E into R determines almost totally the structure of E as a Banach space. In the context of homomorphisms on Banach lattices, we should highlight the recent paper [34], where a James type theorem was proved for positive linear functionals on some Banach lattices (see [34, §6]). For classical Banach lattices (as c0, Lp(μ)-, and C(K)-spaces), the set Hom(X, R) is very small, in the sense that, not just a James type theorem does not hold, and that every homomorphism attains its norm. We characterize lattice homomorphisms attaining their norm on F BL[E] whenever E is an isometric predual of 1(A) or is isometric to 1(A) for some infinite set A These results allow us to show that no Bishop-Phelps theorem holds in the class of Banach lattices, i.e. that there are lattice homomorphisms which cannot be approximated in norm by norm-attaining lattice homomorphisms
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