Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}\_1$ and let $\mathbb{F}\_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}\_n$ on $\mathcal{R}$.


Does the von Neumann crossed product $\mathcal{R}\rtimes\_{\alpha}\mathbb{F}\_n$ have the QWEP?


Remarks: Since $\mathbb{F}\_n$ is a residually finite group, the group von Neumann algebra $\rm{VN}(\mathbb{F}\_n)$ has the QWEP. Moreover $\mathcal{R}$ has the QWEP.



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 $\mathbb F\_n$.


