  In this paper we study the non-amenability question of the Thompson group $F$
from the $C^{*}$ algebra side. Using a characterization of amenability in this
framework we set about evaluating the reduced norm of the averages
$\frac{1}{n}\sum x_0^ix_1x_0^{-i}$, where $x_0$ and $x_1$ are the generators of
$F$ in its finite presentation. We prove that when $n$ is sufficiently large
the above norm concentrates on a specific subset of $F$, easy to describe using
the new normal form for elements in $F$, found by Guba and Sapir. We view this
subset as the only obstruction against non-amenability.
