Multipliers and lacunary sets in non-amenable groups

Let $G$ be a discrete group. Let $\lambda : G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of $\{\lambda(x),x \in G\}$ by $$\sum_{x \in G} f(x) \lambda(x) \to \sum_{x \in G} f(x) \ph(x) \lambda(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_\lambda^*(G)$ which is generated by $\lambda$ ($C_\lambda^*(G)$ is the closure of $K(G)$ in $B(\ell_2(G),\ell_2(G))$.) One of our main results gives a simple characterization of the functions $\ph$ such that $\eps \ph$ is a c.b. multiplier on $C_\lambda^*(G)$ for any bounded function $\eps$, or equivalently for any choice of signs $\eps(x) = \pm 1$. We also consider the case when this holds for ``almost all" choices of signs.