We introduce the notion of stationary actions in the context of C*-algebras. As an application of this concept we prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic and conceptual understanding of why C*-simplicity is stronger than the unique trace property. In addition it allows us to conclude C*-simplicity of new classes of examples, including recurrent subgroups of C*-simple groups.
This is joint work with Yair Hartman.