Norm-attaining lattice homomorphisms and renormings of Banach lattices
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910827721785344 |
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| author | Bilokopytov, Eugene García-Sánchez, Enrique de Hevia, David Martínez-Cervantes, Gonzalo Tradacete, Pedro |
| author_facet | Bilokopytov, Eugene García-Sánchez, Enrique de Hevia, David Martínez-Cervantes, Gonzalo Tradacete, Pedro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Norm-attaining lattice homomorphisms and renormings of Banach lattices Bilokopytov, Eugene García-Sánchez, Enrique de Hevia, David Martínez-Cervantes, Gonzalo Tradacete, Pedro Functional Analysis 46B20, 46B42, 46B04 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. |
| title | Norm-attaining lattice homomorphisms and renormings of Banach lattices |
| topic | Functional Analysis 46B20, 46B42, 46B04 |
| url | https://arxiv.org/abs/2502.10165 |