Selfless reduced $C^{*}$-algebras of linear groups

TL;DR

This paper proves that the reduced C*-algebra of certain linear groups with trivial amenable radical is selfless, establishing a connection between selflessness and simplicity in these algebras, and extends results to twisted versions.

Contribution

It demonstrates that for nontrivial linear groups with trivial amenable radical, their reduced C*-algebras are selfless, linking selflessness with simplicity and extending to twisted cases.

Findings

Reduced C*-algebras of these groups are selfless.

Selflessness and simplicity are equivalent for these algebras.

Results apply to twisted reduced group C*-algebras.

Abstract

It is shown that the reduced C*-algebra of a nontrivial linear group $\Gamma<GL_{d}(k)$ with trivial amenable radical is selfless. Thus selflessness and simplicity coincide for reduced C*-algebras of linear groups. Similar results are obtained for twisted reduced group C*-algebra.