Then there are $*$-homomorphisms $\Phi\colon\LocAlg(H_X)^G \to\LocAlg(H_{X/G})$ and $\Psi\colon\LocAlg(H_{X/G})\to\LocAlg(H_X)^G$ which give mutually inverse $*$-isomorphisms on the quotients
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@@ -1163,6 +1163,15 @@ and we say, that $G$ satisfies the Baum--Connes conjecture, if this map is an is
\todo[inline]{Should contain the following: Rips complexes, Classifying space variants for BCC for $G$, BCC for Euclidean space}
\begin{theorem}
Assume $G$ is finitely generated and torsion free and admits a classifying space $\B G$ which is a finite CW complex.
Then the Baum--Connes assembly map for $G$ (acting on itself) identifies with the composition