Proximality and selflessness for group C*-algebras

TL;DR

This paper establishes conditions under which certain group C*-algebras are selfless, extending previous results and demonstrating stability of selflessness under tensor products.

Contribution

It generalizes recent findings by identifying new classes of groups with selfless reduced group C*-algebras and proves stability of selflessness in specific C*-algebra contexts.

Findings

Reduced group C*-algebras of certain groups are selfless.

Selflessness is stable under tensor products among exact C*-algebras.

A C*-probability space is selfless if it is simple and purely infinite or simple, exact, Z-stable, and uniquely tracial.

Abstract

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 and all Zariski-dense subgroups of PSL(n,R), are selfless in the sense of L. Robert. This generalizes the recent results of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, and of Vigdorovich. 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.