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 ${\sf B}(\ell^2(\Gamma))$.