oa.operator algebras - Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?

Yes. If $a$ and $b$ are generators of $mathbb F_2$ then $mathcal R rtimes_alpha mathbb F_2$ decomposes as an amalgamated free product of $(mathcal R rtimes_alpha langle a rangle)$ and $(mathcal R rtimes_alpha langle b rangle)$ over $mathcal R$, where each of these are hyperfinite. Brown, Dykema, and Jung showed in http://arxiv.org/abs/math/0609080 that for separable finite von Neumann algebras being embeddable into $mathcal R^omega$ is stable under amalgamated free products over a hyperfinite von Neumann algebra. Thus $mathcal R rtimes_alpha mathbb F_2$ is embeddable into $mathcal R^omega$, which is equivalent to QWEP. Induction then gives the case when $2 leq n < infty$, and the case $n = infty$ then follows since QWEP is preserved under (the weak-closure of) increasing unions.

Related to this, Collins and Dykema in http://arxiv.org/abs/1003.1675 have recently shown that the class of Sophic groups is stable under taking amalgamated free products over amenable groups.

I believe this is an open problem however if we consider arbitrary residually finite groups instead of only $mathbb F_n$.