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?



Is it possible to extend this homomorphism?
No. Take $\alpha=g$, a group element, and consider a non-trivial one dimensional representation of the cyclic group generated by $g$. If $G$ has no abelian quotient then you're doomed.


