Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

TL;DR

This paper develops a nonlinear geometric theory for operator spaces by introducing Lipschitz embeddability and Lipschitz-free operator spaces, revealing new structural restrictions and properties in the operator space setting.

Contribution

It introduces the operator space version of Lipschitz-free Banach spaces and explores Lipschitz embeddability, establishing a nonlinear geometric framework for operator spaces.

Findings

Lipschitz embeddability is weaker than linear embeddability but still imposes linear restrictions.

Separable operator spaces satisfy a Lipschitz-lifting property.

Gateaux differentiability of Lipschitz maps is studied in the operator space context.

Abstract

We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space structures. This shows that there is a nontrivial theory of nonlinear geometry for operator spaces and it answers a question in [BCD21]. For that, we introduce the operator space version of Lipschitz-free Banach spaces and prove several properties of it. In particular, we show that separable operator spaces satisfy a sort of isometric Lipschitz-lifting property in the sense of G. Godefroy and N. Kalton. Gateaux differentiability of Lipschitz maps in the operator space category is also studied.