On the small scale nonlinear theory of operator spaces

TL;DR

This paper explores the small scale geometry of operator spaces, revealing that many maps are non-linear yet Lipschitz in small scales, and introduces a geometric parameter that constrains such maps.

Contribution

It generalizes previous results on completely coarse maps to small scale settings and introduces a new geometric parameter for Hilbertian operator spaces.

Findings

Many non-linear maps are completely Lipschitz in small scale.

A geometric parameter restricts the existence of certain maps.

Extension of linearity results to small scale geometries.

Abstract

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.