  In this paper we prove that the Thompson groups $T$ and $V$ are not inner
amenable. In particular, their group von Neumann algebras do not have property
$\Gamma$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$
is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a
few new equivalent characterizations of amenability of $F$.
