Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha=\sum a\_gg$ in $\mathbb C[G]$ (complex functions on $G$ with compact support), $\alpha^\*$ is defined to be $\sum\bar{a\_g}g^{-1}$ and for $\beta=\sum b\_gg\in\mathbb C[G]$, we have the (convolution) product $\alpha\beta:=\sum a\_gb\_hgh$. Let $\alpha=\alpha^\*$ be an element in $\mathbb C[G]$, define $F(\alpha):=a\_e-\sum\_{g\neq e}|a\_g|$. Consider the sequence $(A\_n)$ defined by 
$$A\_n:=F\left(\alpha^{2n}\right)\quad(n\in\mathbb N)$$
Does the sequence $(A\_n)$ contain a nonnegative number?



Does this sequence contain a nonnegative number?
No in general. If $G$ is an infinite cyclic group generated by $g$, and $\beta=1 +g+1/g$, the sum of coefficients of $\beta^{2 n}$ equals $9^n$, and the coefficient of 1 is $9^n$ times the probability that a symmetric random walk with steps $0,\pm 1$ ends up in the origin after $2n$ steps. This probability tends to 0. Thus certain large power $\alpha=\beta^N$ is a counterexample. 


