Recently in my investigations I faced a problem related to amenable groups. I have no idea if my question is suitable for this site, but after ask some collegues in my department I decided to try to find some help here.

Suppose that $G$ is an amenable group having a mean $ m\in \left(L^{\infty}(G,\mathbb{R}) \right)^{*} $. Is it true that there exist a mean $\tilde{m}\in \left(L^{\infty}(G\times G,\mathbb{R}) \right)^*$ such that for all borelians $B$ we have $$ \tilde{m}(B\times G)=m(B). $$

If the answer is negative could you point me out the counter-example ?

Remark: The answer is positive if $G$ is locally compact group because the mean in $G\times G$ is the product Haar measure of $G$.