Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and $\Delta$-points
| Autor: | Cobollo, Christian, Isert, Daniel, López-Pérez, Ginés, Martín, Miguel, Perreau, Yoël, Quero, Alicia, Quilis, Andrés, Rodríguez-Vidanes, Daniel L., Zoca, Abraham Rueda |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\Delta$-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 $\Delta$-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. Comment: 16 pages |
| Databáze: | arXiv |
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