Projective tensor products where every element is norm-attaining
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913431406247936 |
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| author | García-Lirola, Luis C. Guerrero-Viu, Juan Zoca, Abraham Rueda |
| author_facet | García-Lirola, Luis C. Guerrero-Viu, Juan Zoca, Abraham Rueda |
| contents | In this paper we analyse when every element of $X\widehat{\otimes}_πY$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ space and $Y$ is $1$-complemented in its bidual under approximation properties assumptions. This result allows us to provide some new examples where $X$ is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in $X\widehat{\otimes}_πY$ if, for instance, $X=L_1(μ)$ and $Y$ is any Banach space, or if $X$ has the metric $π$-property and $Y$ is a dual space with the RNP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective tensor products where every element is norm-attaining García-Lirola, Luis C. Guerrero-Viu, Juan Zoca, Abraham Rueda Functional Analysis 46B04, 46B20, 46B28 In this paper we analyse when every element of $X\widehat{\otimes}_πY$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ space and $Y$ is $1$-complemented in its bidual under approximation properties assumptions. This result allows us to provide some new examples where $X$ is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in $X\widehat{\otimes}_πY$ if, for instance, $X=L_1(μ)$ and $Y$ is any Banach space, or if $X$ has the metric $π$-property and $Y$ is a dual space with the RNP. |
| title | Projective tensor products where every element is norm-attaining |
| topic | Functional Analysis 46B04, 46B20, 46B28 |
| url | https://arxiv.org/abs/2407.10710 |