Operator space theory from the maximal viewpoint

Abstract

We initiate a theory of ``operator spaces" (in other words subspaces of $C^*$-algebras) with an emphasis on the maximal tensor product. The natural morphisms become the maximally bounded maps instead of the completely bounded ones. With this new looking glass we first examine various issues : lifting and extension properties, local reflexivity, inner derivations, similarity problems.This first part is semi-expository while the second one focuses on the class of von Neumann algebras $M$ such that any completely bounded unital homomorphism on a $C^*$-algebra with values in $M$ is maximally bounded (or equivalently is similar to a $*$-homomorphism via a similarity belonging to $M$). We show that this holds if and only if $M$ is the direct sum of a finite von Neumann algebra and an injective one. This also characterizes the $M$'s such that all completely bounded derivations into $M$ are inner. Our main tool for this is a new characterization of injective von Neumann algebras.