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 $E\_G$.



existence of charaterization of amenable groups by complementation?
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)...


