Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909172781547520 |
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| author | Haller, Rainis Kaasik, Jaan Kristjan Ostrak, Andre |
| author_facet | Haller, Rainis Kaasik, Jaan Kristjan Ostrak, Andre |
| contents | We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter two properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter two property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter two property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions Haller, Rainis Kaasik, Jaan Kristjan Ostrak, Andre Functional Analysis Primary 46B04, Secondary 46B20 We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter two properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter two property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter two property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones. |
| title | Separating diameter two properties from their weak-star counterparts in spaces of Lipschitz functions |
| topic | Functional Analysis Primary 46B04, Secondary 46B20 |
| url | https://arxiv.org/abs/2404.11430 |