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Norm-attaining lattice homomorphisms and renormings of Banach lattices

Eugene Bilokopytov, Enrique García-Sánchez, David de Hevia, Gonzalo Martínez-Cervantes, Pedro Tradacete

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, Rodríguez Abellán, Rueda Zoca and the fourth author.

Norm-attaining lattice homomorphisms and renormings of Banach lattices

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 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 the fourth author.

Paper Structure

This paper contains 7 sections, 27 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Coordinate functionals on a Banach lattice
  3. Stability of norm attainment of lattice homomorphism under renormings
  4. Norm attainment of lattice homomorphisms on AM-spaces
  5. Lattice homomorphisms on AM-spaces and renormings
  6. Free Banach lattices over Banach spaces
  7. Free Banach lattices over lattices

Key Result

Proposition 2.1

Let $x^*$ be a non-zero lattice homomorphism on a Banach lattice $X$, and let $x_{0}\in X_{+}$ be an atom. The following conditions are equivalent. In this case, $y^*\in X^*$ is disjoint with $x^*$ if and only if $y^*(x_0)=0$.

Theorems & Definitions (67)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • ...and 57 more