Ideal structure of the C*-algebra of Thompson group T

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.