Does this element belong to all powers of the augmentation ideal of the group algebra.

Question

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $b$ such that $\alpha\beta=a\alpha+b\beta$. Does it imply that $a\alpha+b\beta\in I^n$, for all $n\in\mathbb N$?

Answer 1

I think this is not always true. For example, if $\alpha=g-1$ and $\beta=g^{-1}-1$, then $$\alpha\beta=-(\alpha+\beta)$$ and if $\bigcap_{n\in\mathbb N}I^n=\{0\}$, then it is impossible to have $\alpha+\beta\in I^n$, for all $n\in\mathbb N$.