Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that

- $\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent,
- $(\alpha^n)_{n\in\mathbb N}$ converges to 0 in strong operator topology, so in particular $\lim_n\|\alpha^n\|_2=0$.

Assume that $K$ is the closed linear span of $\{1,\alpha,\alpha^2,\dotsc\}$ in $\ell^2(G)$. Is $\{1,\alpha,\alpha^2,\dotsc\}$ a Schauder basis for $K$?