existence of charaterization of amenable groups by complementation?

Question

Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.

Do you know a charaterization of discrete amenable groups by the existence of a complementation of a closed space $F$ of a Banach space $E$?

More precisely, the required charaterization is
For all discrete group $G$, there exists a Banach space $E_G$ and a closed space $F_G$ of $E_G$ such that $G$ is amenable if and only if $F_G$ is complemented in $E_G$.

Answer 1

Yes: a discrete group $G$ is amenable if and only if the reduced group C*-algebra $C^*_r(G)$ is nuclear, see E.C. Lance, On nuclear $C^{\ast} $-algebras. J. Functional Analysis 12 (1973), 157--176. This is then equivalent to $W^*(G) = C^*_r(G)^{**}$ being an injective von Neumann algebra: which by definition means that if $W^*(G) \subseteq B(H)$ then there is a contractive projection from $W^*(G)$ to $B(H)$.

I'm pretty sure you could look at the group von Neumann algebra $VN(G)$ instead, but I cannot recall the correct reference (but it's all in Runde's book "Lectures on Amenability"). Note that all this only works because $G$ is discrete.

Now, the problem is that you do need ``contractive'' projection here: it's still a conjecture if just having a bounded projection is enough.

Also, I'm sure there are other answers (and perhaps some that are easier: even a streamlined approach to all this uses a lot of operator algebra theory)...