# existence of charaterization of amenable groups by complementation?

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 $E_G$.

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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)...

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Thank you. Other charaterizations are welcome. –  BigBill Oct 3 '10 at 8:51
I won't have accepted this! Indeed, I was sort of hoping someone else might come along and give a characterisation which didn't need "contractive" projection... –  Matthew Daws Oct 3 '10 at 9:07
Dear Matthew, here's an ad hoc supplement to your argument. Although it is not known whether complementability of a von Neumann subalgebra $M \subset B(H)$ by a bounded projection implies injectivity, it is true for properly infinite von Neumann algebras. So, $vN(G) \bar{\otimes} B(\ell_2) \subset B(\ell_2(G) \otimes \ell_2)$ is complemented by a bounded projection if and only if $G$ is amenable (assuming $G$ is discrete). –  Narutaka OZAWA Oct 4 '10 at 11:25
Ah! Very nice! –  Matthew Daws Oct 4 '10 at 12:11