Y 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 Id(M) = vu : M(sic)B(H)(sic)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).

Automated summary

It is shown that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in a certain sense. However, there are examples of algebras that are not seemingly injective, such as B(H)** and certain finite examples defined using ultraproducts. Moreover, it suffices for an algebra to be seemingly injective to have a specific factorization of the identity through B(H) with certain positive properties.