Does it true that for every group $G$ and $f\in C[G]$ it holds that $dim(C[G]*f)\mathop{supp}(f)\geq |G|$? Where $C[G]$ is the group algebra, and by $C[G]*f$ I mean left ideal of the group algebra $C[G]$ generated by $f$.

Essentially this is uncertainty principle for non-commutative groups. Since $supp \hat {f} = dim C[G]*f$.