Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911763633537024 |
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| author | Cobollo, Christian Isert, Daniel López-Pérez, Ginés Martín, Miguel Perreau, Yoël Quero, Alicia Quilis, Andrés Rodríguez-Vidanes, Daniel L. Zoca, Abraham Rueda |
| author_facet | Cobollo, Christian Isert, Daniel López-Pérez, Ginés Martín, Miguel Perreau, Yoël Quero, Alicia Quilis, Andrés Rodríguez-Vidanes, Daniel L. Zoca, Abraham Rueda |
| contents | We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is weakly dense; (3) The set of ccw $Δ$-points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming.
We also show that there are points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $Δ$-points, meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03610 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points Cobollo, Christian Isert, Daniel López-Pérez, Ginés Martín, Miguel Perreau, Yoël Quero, Alicia Quilis, Andrés Rodríguez-Vidanes, Daniel L. Zoca, Abraham Rueda Functional Analysis We prove that there exists an equivalent norm $\Vert\vert\cdot\vert\Vert$ on $L_\infty[0,1]$ with the following properties: (1) The unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is weakly dense; (3) The set of ccw $Δ$-points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ is norming. We also show that there are points of the unit ball of $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ which are not $Δ$-points, meaning that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ fails the diametral local diameter 2 property. Finally, we observe that the space $(L_\infty[0,1],\Vert\vert\cdot\vert\Vert)$ provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces. |
| title | Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $Δ$-points |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2309.03610 |