On the small scale nonlinear theory of operator spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916229451612160 |
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| author | Braga, Bruno de Mendonça Chávez-Domínguez, Javier Alejandro |
| author_facet | Braga, Bruno de Mendonça Chávez-Domínguez, Javier Alejandro |
| contents | We initiate the study of the small scale geometry of operator spaces. The authors have previously shown that a map between operator spaces which is completely coarse (that is, the sequence of its amplifications is equi-coarse) must be $\mathbb R$-linear. We obtain a generalization of the aforementioned result to completely coarse maps defined on the unit ball of an operator space. By relaxing the condition to a small scale one, we prove that there are many non-linear examples of maps which are completely Lipshitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the small scale nonlinear theory of operator spaces Braga, Bruno de Mendonça Chávez-Domínguez, Javier Alejandro Operator Algebras Functional Analysis We initiate the study of the small scale geometry of operator spaces. The authors have previously shown that a map between operator spaces which is completely coarse (that is, the sequence of its amplifications is equi-coarse) must be $\mathbb R$-linear. We obtain a generalization of the aforementioned result to completely coarse maps defined on the unit ball of an operator space. By relaxing the condition to a small scale one, we prove that there are many non-linear examples of maps which are completely Lipshitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps. |
| title | On the small scale nonlinear theory of operator spaces |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2404.19092 |