Let $g$ be a group element, let $G$ be the group generated by $g$, and let $\mathbb C G$ be the group algebra on $G$.

If we define $\tau:G\to\mathbb C$ as \begin{align*}\tag{1} \tau[g^n]=\begin{cases} 1&\text{if }n=0;\text{ and}\\ \alpha_n&\text{otherwise}, \end{cases} \end{align*} where $(\alpha_n:n\in\mathbb N)\subset\mathbb C$ are completely arbitrary, then $\tau$ can be extended to a normalized (i.e., $\tau[1]=1$) linear functional on $\mathbb C G$.

If, however, we would like $\tau$ to be positive (i.e., $\tau[pp^*]\geq0$ for all $p\in\mathbb C G$), there are certain restrictions on how we can choose the $\alpha_n$ in $(1)$ (for instance, we have to make sure that $\tau[g^n]=\overline{\tau[g^{-n}]}$ for all $n$).

In this context, the question I'm interested in is the following:

Question.How many of the $\alpha_n$ can be different from zero (and under what conditions)?

If $g$ has finite order $m\in\mathbb N$, then we can choose as many of $\alpha_1,\ldots,\alpha_{\lfloor m/2\rfloor}$ nonzero as we want, provided $|\alpha_1|,\ldots,|\alpha_{\lfloor m/2\rfloor}|$ are bounded above by a small enough quantity (this is mainly because, in this case, any element of $\mathbb C G$ can be written as $\beta_0+\beta_1g+\beta_2 g^2+\cdots+\beta_{m-1}g^{m-1}$).

However, if $g$ is of infinite order, I can't find an example of a positive $\tau$ with at least one $\alpha_n$ nonzero, but I can't show that all of the $\alpha_n$ need to be zero either. Is there a known characterization of positive linear functionals on such algebras?