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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.

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By the way, it's good practice to define your notation (QWEP?). You wouldn't want to make the mistake of Serre, "Let X be an E.C. with. C.M." where "with." stands, of course, for "without". =) – Harry Gindi Jul 10 '10 at 7:52
@Harry: normally I agree, but I think almost everyone in the intended audience for this question would have heard of, or at least know how to interpret Google results for, "QWEP". Not that I have any clue as to the answer, mind you. – Yemon Choi Jul 10 '10 at 9:27
QWEP = Quotient Weak Expectation Property – Jesse Peterson Jul 10 '10 at 9:32
up vote 8 down vote accepted

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 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 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 $\mathbb F_n$.

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I am very grateful to you for showing me all this! A last question: do you know a reference for the following assertion $\mathcal{R}\rtimes_{\alpha} \mathbb{F}_2$ decomposes as an amalgamated free product of $\mathcal{R} \rtimes_{\alpha}<a>$ and $\mathcal{R}\rtimes_{\alpha}<b>$ over $\mathcal{R}$. – BigBill Jul 10 '10 at 13:24
This follows, more or less, directly from the definition of amalgamated free products for tracial von Neumann algebras. You just need to show that $\mathcal R \rtimes_\alpha \langle a \rangle$ and $\mathcal R \rtimes_\alpha \langle b \rangle$ are freely independent relative to $\mathcal R$. See for instance – Jesse Peterson Jul 10 '10 at 18:22
Thank you very much. – BigBill Jul 13 '10 at 14:54

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