Let $G$ be non-amenable finitely generated group.
1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that
$\frac{1}{|S(n)|}||\sum_{g\in S(n)} \lambda(g)||\rightarrow 0$ when $n\rightarrow \infty$.
2) The same as (1), but $S(n)$ is finite subset of $G$.
Here $\lambda:G\rightarrow B(l^2(G))$ is left regular representation of $G$.
Also 1) is reformulation of 2).
Edit: here are some discussions on the question.