Proximality and selflessness for group C*-algebras

We prove that the reduced group C*-algebras of infinite countable discrete groups having topologically-free extreme boundaries, or more generally groups that satisfy certain combinatorial property including all acylindrically hyperbolic groups with no nontrivial finite normal subgroups, are selfless in the sense of L. Robert. This generalizes the recent result of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell. We also prove that selflessness is stable under tensor product among exact C*-algebras and that a C*-probability space is selfless provided that it is either simple and purely infinite or simple, exact, Z-stable, and uniquely tracial.