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 $E_G$.