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 M I d M = v u : M → u B ( H ) → v 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 (ℓ 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).