Functoriality of the reduced assembly map

Determine whether the reduced Baum–Connes assembly map μ_G is functorial for arbitrary group homomorphisms, rather than only for injective homomorphisms.

Background

The full assembly map is fully functorial because full group C*-algebras are functorial for all group homomorphisms. In contrast, reduced group C*-algebras are generally functorial only for injective homomorphisms, leaving functoriality of the reduced assembly map unresolved for arbitrary homomorphisms.

References

It is not known whether $\mu_G$ is functorial. However, since the reduced group $C*$-algebra is functorial for injective group homomorphisms (every injective homomorphism $G\hookrightarrow H$ induces an injective $$-homomorphism $C^_r(G) \hookrightarrow C*_r(H)$), the map $\mu_G$ is certainly functorial with respect to injective group homomorphisms.

— The Baum-Connes conjecture: a concise course  (2610.01802 - Valette, 1 Oct 2026) in Section 7.3.2, “Functoriality of assembly”