Banach spaces & Banach lattices Seminar
BANACH SPACES & BANACH LATTICES SEMINAR - Norm-attaining lattice homomorphisms
Speaker: David de Hevia (ICMAT)Date: Thursday, 06 February 2025 - 12:30Place: Aula Gris 1, ICMAT
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 talk we study Banach lattices on which every 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 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, Rodríguez-Abellán, Rueda Zoca and Martínez-Cervantes.
