Seemingly injective von Neumann algebras

TL;DR

This paper characterizes certain von Neumann algebras with a positive approximation property as being 'seemingly injective,' revealing that many such algebras are isomorphic to B(ell_2), but some are not due to the failure of the approximation property.

Contribution

It introduces the concept of 'seemingly injective' von Neumann algebras and characterizes them via a specific factorization involving positive maps, linking this to the weak* positive approximation property.

Findings

Separable predual von Neumann algebras are isomorphic to B(ell_2).

Certain free group von Neumann algebras are seemingly injective.

Some algebras like B(H)** are not seemingly injective due to the failure of the approximation property.

Abstract

We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu: M{\buildrel u\over\longrightarrow} B(H) {\buildrel v\over\longrightarrow} M$$ with $u$ normal, unital, positive and $v$ completely contractive. As a corollary, if $M$ has a separable predual, $M$ is isomorphic (as a Banach space) to $B(\ell_2)$. For instance this applies (rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since $B(H)$ fails the approximation property (due to Szankowski) there are $M$'s (namely $B(H)^{**}$ and certain finite examples defined using ultraproducts) that are not seemingly injective. Moreover, for $M$ to be seemingly injective it suffices to have the above factorization of $Id_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal).