We explore the ideal structure of the reduced C∗-algebra of R. Thompson’s group T. We show that even though T has trace, one cannot use the Kesten Condition to verify that the reduced C∗-algebra of T is simple. At the time of the initial writing of this chapter, there had been no example group for which it was known that the Kesten Condition would fail to prove simplicity, even though the group has trace. Motivated by this first result, we describe a class of groups where even if the group has trace, one cannot apply the Kesten Condition to verify the simplicity of those groups' reduced C∗-algebras. We also offer an apparently weaker condition to test for the simplicity of a group's reduced C∗-algebra, and we show this new test is still insufficient to show that the reduced C∗-algebra of T is simple. Separately, we find a controlled version of a Ping-Pong Lemma which allows one to find non-abelian free subgroups in groups of homeomorphisms of the circle generated by elements with rational rotation number. We use our Ping-Pong Lemma to find a simple converse to a theorem of Uffe Haagerup and Kristian Knudsen Olesen.