Duality of Lipschitz-free spaces over ultrametric spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917002695671808 |
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| author | Abrahamsen, Trond A. Lima, Vegard Ostrak, Andre |
| author_facet | Abrahamsen, Trond A. Lima, Vegard Ostrak, Andre |
| contents | We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duality of Lipschitz-free spaces over ultrametric spaces Abrahamsen, Trond A. Lima, Vegard Ostrak, Andre Functional Analysis 46B04, 46B10, 46B20, 54E35 We give a metric characterisation of when the Lipschitz-free space over a separable ultrametric space is a dual Banach space. In the case where the Lipschitz-free space has a predual, we show that this predual is M-embedded if and only if the metric space is proper. We show that for ultrametric spaces, the little Lipschitz space is always an M-ideal in the corresponding space of Lipschitz functions, and we show that this is not the case for metric spaces in general, thus answering a question posed by Werner in the negative. Finally, we show that the space of Lipschitz functions of an ultrametric space contains a strongly extreme point. |
| title | Duality of Lipschitz-free spaces over ultrametric spaces |
| topic | Functional Analysis 46B04, 46B10, 46B20, 54E35 |
| url | https://arxiv.org/abs/2509.22328 |