On the small scale nonlinear theory of operator spaces

Braga, Bruno M.; Chávez-Domínguez, Javier Alejandro

Comptes Rendus. Mathématique, Tome 362 (2024) no. G13, pp. 1893-1914

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.

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

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