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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2403.09245 |
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| _version_ | 1866916160046366720 |
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| author | Kurik, Kaarel August |
| author_facet | Kurik, Kaarel August |
| contents | We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by constraining the componentwise behavior of the inverse $g=f^{-1}$ with a theorem admitting a graph-theoretic interpretation. We also show that if $X, Y$ are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X \to B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type $B_{X'} \to B_{X'}$ for some Banach space $X'$, and that a homeomorphic 1-Lipschitz non-isometry of type $B_X \to B_X$ restricts to a homeomorphic 1-Lipschitz non-isometry of type $B_S \to B_S$ for some separable subspace $S \leq X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditional plasticity of the unit ball of the $\ell_\infty$-sum of finitely many strictly convex Banach spaces Kurik, Kaarel August Functional Analysis 46B20, 47H09, 05C69 We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by constraining the componentwise behavior of the inverse $g=f^{-1}$ with a theorem admitting a graph-theoretic interpretation. We also show that if $X, Y$ are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X \to B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type $B_{X'} \to B_{X'}$ for some Banach space $X'$, and that a homeomorphic 1-Lipschitz non-isometry of type $B_X \to B_X$ restricts to a homeomorphic 1-Lipschitz non-isometry of type $B_S \to B_S$ for some separable subspace $S \leq X$. |
| title | Conditional plasticity of the unit ball of the $\ell_\infty$-sum of finitely many strictly convex Banach spaces |
| topic | Functional Analysis 46B20, 47H09, 05C69 |
| url | https://arxiv.org/abs/2403.09245 |