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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2402.13416 |
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| _version_ | 1866916133154586624 |
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| author | Guterman, Alexander Kuzma, Bojan Singla, Sushil Zhilina, Svetlana |
| author_facet | Guterman, Alexander Kuzma, Bojan Singla, Sushil Zhilina, Svetlana |
| contents | For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $Γ(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $Γ_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $Γ(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $Γ_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $Γ_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Birkhoff-James classification of norm's properties Guterman, Alexander Kuzma, Bojan Singla, Sushil Zhilina, Svetlana Functional Analysis 46B20, 05C63 For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $Γ(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $Γ_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $Γ(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $Γ_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $Γ_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes. |
| title | Birkhoff-James classification of norm's properties |
| topic | Functional Analysis 46B20, 05C63 |
| url | https://arxiv.org/abs/2402.13416 |