Baum–Connes conjecture (reduced assembly map isomorphism) for transformation groupoids

Show that the reduced Baum–Connes assembly maps for the transformation groupoid X ⋊ Γ are isomorphisms, i.e., prove that for i in {0,1} the maps from equivariant K-homology to K-theory of the reduced crossed product C*-algebra are isomorphisms.

Background

The paper recalls the classical Baum–Connes conjecture for actions of countable discrete groups on compact spaces. In this setting, the conjecture asserts that the reduced assembly maps are isomorphisms. While known for many classes (e.g., a-T-menable groups), counterexamples to exactness led to failures in certain groupoid contexts, motivating rectified versions of the conjecture.

References

In , see also , the reduced BC-maps were conjectured to be isomorphisms.

— Admissible Higson-Roe sequences for transformation groupoids  (2411.00182 - Benameur et al., 2024) in Subsection 4.1 (Review of the BC assembly map)

(the {\bf Baum-Connes conjecture}) For $i=0,1$, the assembly map \begin{equation}\label{BCstandard} \mu_G: K_iG(\underline{E}G)\rightarrow K_i(C*_r(G)) \end{equation} is an isomorphism.

BCstandard:

μG:KiG(E‾G)→Ki(Cr∗(G))\mu_G: K_i^G(\underline{E}G)\rightarrow K_i(C^*_r(G))

— The Baum-Connes conjecture: a concise course  (2610.01802 - Valette, 1 Oct 2026) in Conjecture 1, Section 2, “What is the Baum-Connes conjecture?”

For every $C*$-algebra $B$ equipped with a continuous $G$-action by $*$-automorphisms, the map $\mu_{G,B}:KK_iG(\underline{E}G,B) K_i(B\rtimes_r G)\;(i=0,1)$ is an isomorphism.

— The Baum-Connes conjecture: a concise course  (2610.01802 - Valette, 1 Oct 2026) in Conjecture 5, Section 7.2, “The Baum-Connes conjecture with coefficients”