Seemingly Injective Von Neumann Algebras

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 MIdM=vu:M→uB(H)→vM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I{d_M} = vu:M\buildrel u \over \longrightarrow B(H)\buildrel v \over \longrightarrow M$$\end{document} 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(ℓ2). 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 IdM through B(H) with u, v positive (and u still normal).