Simplicity and Selflessness of Reduced Group C*-Algebras

There are numerous sufficient conditions for the reduced group C*-algebra of a discrete group to be simple, including growth conditions, paradoxical decompositions, and existence of boundary actions. Recently, an important strengthening of C*-simplicity, namely C*-selflessness, has been described and there is a substantial overlap between the techniques used to prove C*-simplicity and C*-selflessness. However, although a characterization of C*-simplicity was found by Kalantar-Kennedy in 2014, no such characterization of C*-selflessness is yet known. I will survey three different approaches taken to prove C*-selflessness and the limitations of each approach.