  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).
