Group-theoretic characterization of pureness

Characterize the discrete groups G for which the reduced group C*-algebra C*_r(G) is pure.

Background

The paper constructs, for every countably infinite discrete group Γ, a nonamenable wreath product whose reduced group C*-algebra is not pure. This provides a counterexample to the converse of the fact that pureness of C*_r(G) implies nonamenability of G.

The authors leave unresolved the broader classification problem: determining precisely which discrete groups have pure reduced group C*-algebras.

References

We conclude by posing the following natural questions.

Is there a group-theoretic characterisation of those discrete groups $G$ for which $C*_r(G)$ is pure?

— Nonamenable groups whose reduced group C*-algebras are not pure  (2609.10653 - Bell, 9 Sep 2026) in Section 2, immediately after the proof of Theorem A

(Kaplansky-Kadison, around 1954) If $G$ is torsion-free, then the only idempotents in $C*_r(G)$ are 0 and 1.

— The Baum-Connes conjecture: a concise course  (2610.01802 - Valette, 1 Oct 2026) in Conjecture 2, Section 3.1, “The Kaplansky-Kadison conjecture”