Is it possible to construct series of groups $G_i, |G_i|\mapsto \infty$ and functions $f_i:G_i\mapsto$ {$ 1,0,-1$} such that $f_i(1)=0$, $f_i(g) \in ${$-1,1$} for $g\neq 1$ such that dimension of $C[G_i]*f_i$ is small (i.e. $\dim C[G_i]*f_i\leq O(|G_i|^{\varepsilon })$ for every $\varepsilon>0$).