C*-selflessness of all C*-simple groups

Determine whether every C*-simple discrete group is C*-selfless.

Background

The paper studies selflessness for reduced group C*-algebras. C*-selflessness implies C*-simplicity, but the converse is not established in general. Previous results prove selflessness for several important classes, including certain acylindrically hyperbolic groups and groups admitting suitable minimal, extremely proximal, and topologically free actions.

The unresolved question asks whether C*-simplicity alone suffices to imply C*-selflessness for every discrete group. The paper does not resolve this general problem; instead, it proves C*-selflessness for the separate class of vigorous groups.