On the small scale nonlinear theory of operator spaces

Bruno M. Braga; Javier Alejandro Chávez-Domínguez

doi:10.5802/crmath.678

Research article - Operator theory

On the small scale nonlinear theory of operator spaces

Bruno M. Braga ¹ ; Javier Alejandro Chávez-Domínguez ²

¹ IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
² Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1893-1914.

Abstract
Résumé

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 $ℝ$ -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.

Nous commençons l’étude de la géométrie à petite échelle des espaces d’opérateurs. Les auteurs ont précédemment montré qu’une application entre espaces d’opérateurs qui est complètement grossière (c’est-à-dire que la séquence de ses amplifications est équi-grossière) doit être $ℝ$ -linéaire. Nous obtenons une généralisation du résultat susmentionné aux applications complètement grossières définies sur la boule unité d’un espace d’opérateurs. En assouplissant la condition à une petite échelle, nous prouvons qu’il existe de nombreux exemples non linéaires d’applications qui sont complètement Lipschitz à petite échelle. Nous définissons un paramètre géométrique pour les espaces d’opérateurs hilbertiens homogènes qui impose des restrictions sur l’existence de telles applications.

Received: 2024-04-29
Revised: 2024-07-19
Accepted: 2024-08-14
Published online: 2024-12-03

DOI: 10.5802/crmath.678

Classification: 47L25, 46L07, 46B80
Keywords: Operator spaces, Coarse geometry, Embeddings
Mots-clés : Espaces d’opérateurs, Géométrie grossière, Injections

Author's affiliations:

Bruno M. Braga ¹; Javier Alejandro Chávez-Domínguez ²

¹ IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
² Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Bruno M. Braga and Javier Alejandro Ch\'avez-Dom{\'\i}nguez},
     title = {On the small scale nonlinear theory of operator spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1893--1914},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.678},
     language = {en},
}

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Bruno M. Braga; Javier Alejandro Chávez-Domínguez. On the small scale nonlinear theory of operator spaces. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1893-1914. doi : 10.5802/crmath.678. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.678/

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