  In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for
Richard Thompson's group $T$, if there exists a finite set $H$ which can be
decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in
H_1}\pi(g)=\sum_{h\in H_2}\pi(h)$ and such that the closed ideal generated by
$\sum_{g\in H_1}\lambda(g)-\sum_{h\in H_2}\lambda(h)$ coincides with
$C^*_\lambda(T)$, then the Richard Thompson group $F$ is not amenable. In
particular, if $C_{\lambda}^*(T)$ is simple then $F$ is not amenable. Here we
prove the converse, namely, if $F$ is not amenable then we can find two sets
$H_1$ and $H_2$ with the above properties. The only currently available tool
for proving simplicity of group $C^*$-algebra is Power's condition. We show
that it fails for $C_{\lambda}^*(T)$ and present an apparent weakening of that
condition which could potentially be used for various new groups $H$ to show
the simplicity of $C_{\lambda}^*(H)$. While we use our weakening in the proof
of the first result, we also show that the new condition is still too strong to
be used to show the simplicity of $C_{\lambda}^*(T)$. Along the way, we give a
new application of the Ping-Pong Lemma to find free groups as subgroups in
groups of homeomorphisms of the circle generated by elements with rational
rotation number.
