$C^{\ast }$ Estimates for Averaging Sums of Elements in the Thompson Group $F$

Abstract

In this paper we study the non-amenability question of the Thompson group $F$ from the $C^{\ast }$ algebra side. Using a characterization of amenability in this framework we set about evaluating the reduced norm of the averages $\frac{1}{n}\sum x_0^i{x}_1{x}_0^{-i}$, 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.

submitted to the Proceedings of the OAMP3 conference, Bucharest, August, 2005