Diversity of Lipschitz-free spaces over countable complete discrete metric spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909623348363264 |
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| author | Basset, Estelle Lancien, Gilles Procházka, Antonín |
| author_facet | Basset, Estelle Lancien, Gilles Procházka, Antonín |
| contents | We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index $D$ presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of $\{D(\mathcal F(M)): M$ countable, complete, discrete$\}$ is uncountable while $\{D(\mathcal F(M)):M$ infinite, compact, purely 1-unrectifiable$\}=\{ω,ω^2\}$. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: $\{D(\mathcal F(M)):M$ infinite, uniformly discrete$\}=\{ω^2,ω^3\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diversity of Lipschitz-free spaces over countable complete discrete metric spaces Basset, Estelle Lancien, Gilles Procházka, Antonín Functional Analysis Metric Geometry 46B20, 46B22, 51F30, 30L05, 54E45 We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index $D$ presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of $\{D(\mathcal F(M)): M$ countable, complete, discrete$\}$ is uncountable while $\{D(\mathcal F(M)):M$ infinite, compact, purely 1-unrectifiable$\}=\{ω,ω^2\}$. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: $\{D(\mathcal F(M)):M$ infinite, uniformly discrete$\}=\{ω^2,ω^3\}$. |
| title | Diversity of Lipschitz-free spaces over countable complete discrete metric spaces |
| topic | Functional Analysis Metric Geometry 46B20, 46B22, 51F30, 30L05, 54E45 |
| url | https://arxiv.org/abs/2505.19891 |