A central limit theorem in the framework of the Thompson group $F$

We discuss a central limit theorem in the framework of the group algebra of the Thompson group $F$. We consider the sequence of self-adjoint elements given by $a_n=\frac{g_n+g_n^{\ast }}{\sqrt{2}}$ in the noncommutative probability space $(\mathbb{C}(F),\phi )$, where the expectation functional $\phi$ is the trace associated to the left regular representation of $F$, and the $g_n$-s are the generators of $F$ in its standard infinite presentation. We show that the limit law of the sequence $s_n=\frac{a_0+\cdots +a_{n-1}}{\sqrt{n}}$ is the standard normal distribution.