Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space

Question

Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$ and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that $\VN(\Gamma)$ is injective as a von Neumann algebra if and only if $\Gamma$ is amenable.

I'm looking for von Neumann subalgebras of ${\sf B}(\ell^2(\Gamma))$ which are injective and contain $\VN(\Gamma)$. Of course, ${\sf B}(\ell^2(\Gamma))$ is one itself, but does the family of such subalgebras have a least element with respect to inclusion? If not, is there something smaller than ${\sf B}(\ell^2(\Gamma))$? Something that sees the group structure more closely, and which is reasonably canonical? I'd be happy with an answer for $\Gamma={\rm SL}(n, {\bf Z})$, or even for $\Gamma$ the free group on two generators.

The corresponding question for ${\rm C}^*$-algebras seems to have a nice answer, at least for discrete exact groups, in recent work of Kalantar and Kennedy http://arxiv.org/abs/1405.4359 However, it isn't clear to me if their crossed product construction embeds into ${\sf B}(\ell^2(\Gamma))$.

Answer 1

Since a von Neumann algebra is injective if and only if its commutant is injective, this is the same as finding injective von Neumann subalgebras of $VN(\Gamma)' \cong VN(\Gamma)$. Maximal injective subalgebras of $VN(\Gamma)$ have been studied quite a bit. In particular, they always exist and for $\Gamma$ a free group, the von Neumann subalgebra generated by one of the group generators gives an explicit example [Popa: Maximal injective subalgebras in factors associated with free groups. Adv. Math., 50 (1983), 27-48.]. For $\Gamma = SL(n, \mathbb Z)$, it's a recent result that the subgroup of upper triangular matrices generates a maximal injective von Neumann subalgebra [Boutonnet, Carderi: Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups, arXiv:1411.4093].

Related to boundaries, I should mention that if $\mu \in {\rm Prob}(\Gamma)$ then the corresponding Furstenberg boundary $(B, \eta)$ is amenable in Zimmer's sense and hence gives rise to an injective von Neumann algebra $L^\infty(B, \eta) \rtimes \Gamma$ [Zimmer, Hypernite factors and amenable ergodic actions, Invent. Math. 41 (1977), no. 1, 23-31.] This crossed product typically does not embed as an intermediate von Neumann algebra between $VN(\Gamma)$ and $\mathcal B(\ell^2\Gamma)$. ($L^\infty(B, \eta) \rtimes \Gamma$ is typically type $III$ while intermediate von Neumann algebras are always type $I$ or type $II$). However, $L^\infty(B, \eta) \rtimes \Gamma$ does always embed between $VN(\Gamma)$ and $\mathcal B(\ell^2\Gamma)$ as an operator system. You can find details on this in the appendix to [Izumi: E0-semigroups: around and beyond Arveson's work. J. Operator Theory 68 (2012), no. 2, 335–363.]