A Choquet theory of Lipschitz-free spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912301542539264 |
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| author | Smith, Richard J. |
| author_facet | Smith, Richard J. |
| contents | Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $β\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Radon measures on $β\widetilde{M}$ that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the $\preccurlyeq$-minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05177 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Choquet theory of Lipschitz-free spaces Smith, Richard J. Functional Analysis 46B20, 46A55 (Primary) Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $β\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Radon measures on $β\widetilde{M}$ that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the $\preccurlyeq$-minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces. |
| title | A Choquet theory of Lipschitz-free spaces |
| topic | Functional Analysis 46B20, 46A55 (Primary) |
| url | https://arxiv.org/abs/2412.05177 |