Norm-attaining lattice homomorphisms
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Otros documentos de la autoría: Dantas, Sheldon; Martínez Cervantes, Gonzalo; Rodríguez Abellán, José David; Rueda Zoca, Abraham
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Título
Norm-attaining lattice homomorphismsAutoría
Fecha de publicación
2021-07-26Editor
EMS (European Mathematical Society)DOI
10.4171/RMI/1292ISSN
0213-2230Cita bibliográfica
Dantas, S., Martínez-Cervantes, G., Abellán, J. D. R., & Zoca, A. R. (2021). Norm-attaining lattice homomorphisms. Revista Matemática Iberoamericana.Tipo de documento
info:eu-repo/semantics/articleVersión
info:eu-repo/semantics/acceptedVersionPalabras clave / Materias
Resumen
In this paper we study the structure of the set Hom(X,R) of all lattice homomorphisms from a Banach lattice X into R. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological ... [+]
In this paper we study the structure of the set Hom(X,R) of all lattice homomorphisms from a Banach lattice X into 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 c0, Lp- 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. [-]
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Revista Matemática Iberoamericana, 2021Derechos de acceso
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