A Ribe theorem for noncommutative L$_1$-spaces and Lipschitz free operator spaces

TL;DR

This paper extends Ribe's theorem to noncommutative L1-spaces and introduces Lipschitz free operator spaces, advancing the nonlinear theory of operator spaces and their embeddings in the noncommutative setting.

Contribution

It establishes a noncommutative Ribe theorem for preduals of von Neumann algebras and introduces Lipschitz free operator spaces, bridging nonlinear Banach space theory and operator space theory.

Findings

Ribe's theorem has a complete analog for preduals of von Neumann algebras

Introduction of Lipschitz free operator spaces

Results on embeddings and properties of injective von Neumann algebras

Abstract

These notes have the intent to introduce the study of the nonlinear aspects of operator space theory. We investigate some results on the nonlinear theory of Banach spaces which remain valid in the noncommutative case. In particular, we show that Ribe's theorem has a complete analog to preduals of von Neumann algebras and introduce the concept of the Lipschitz free operator space of an operator space. We use those to prove results about injective von Neumann algebras, Pisier's operator space OH, and the existence of complete linear isometric embeddings between operator spaces.