On the small scale nonlinear theory of operator spaces
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 aforementi...
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Académie des sciences
2024-12-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.678/ |
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| author | Braga, Bruno M. Chávez-Domínguez, Javier Alejandro |
| author_facet | Braga, Bruno M. Chávez-Domínguez, Javier Alejandro |
| author_sort | Braga, Bruno M. |
| collection | DOAJ |
| description | 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 Lipschitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps. |
| format | Article |
| id | doaj-art-5ffcf4b2c4fe4b35ba18ebef7e9c2c16 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-5ffcf4b2c4fe4b35ba18ebef7e9c2c162025-02-07T11:27:00ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-12-01362G131893191410.5802/crmath.67810.5802/crmath.678On the small scale nonlinear theory of operator spacesBraga, Bruno M.0https://orcid.org/0000-0002-3456-4002Chávez-Domínguez, Javier Alejandro1https://orcid.org/0000-0001-5061-3612IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, BrazilDepartment of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USAWe 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 Lipschitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.678/Operator spacesCoarse geometryEmbeddings |
| spellingShingle | Braga, Bruno M. Chávez-Domínguez, Javier Alejandro On the small scale nonlinear theory of operator spaces Comptes Rendus. Mathématique Operator spaces Coarse geometry Embeddings |
| title | On the small scale nonlinear theory of operator spaces |
| title_full | On the small scale nonlinear theory of operator spaces |
| title_fullStr | On the small scale nonlinear theory of operator spaces |
| title_full_unstemmed | On the small scale nonlinear theory of operator spaces |
| title_short | On the small scale nonlinear theory of operator spaces |
| title_sort | on the small scale nonlinear theory of operator spaces |
| topic | Operator spaces Coarse geometry Embeddings |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.678/ |
| work_keys_str_mv | AT bragabrunom onthesmallscalenonlineartheoryofoperatorspaces AT chavezdominguezjavieralejandro onthesmallscalenonlineartheoryofoperatorspaces |