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

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.