Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it possible to extend a non-zero representation of $\mathfrak A$ (on a Hilbert space) to all of $\ell^1(G)$? What is the situation if we consider $G$ to be amenable?