I am currently reading
https://arxiv.org/pdf/math/0001094.pdf
In theorem 5.5 there is an isomorphism
$[\mathbb{K}(H_2)\chi(\mathbb{K}(L^2G\otimes H_1)A),\mathbb{K}(G_2\mathbb{N})B]\cong[\chi_s(A),B]_{\mathbb{K}(G_2\mathbb{N})}$
or writing out the notation:
$[\mathbb{K}(H_2)\otimes \chi(\mathbb{K}(L^2(G_2)\otimes H_1),\mathbb{K}(L^2(G_2)\otimes l^2(\mathbb{N}))\otimes B]\cong [\mathbb{K}(L^2(G_2)\otimes l^2(\mathbb{N}))\otimes\chi(\mathbb{K}(L^2(G_2)\otimes l^2(\mathbb{N})\otimes A),\mathbb{K}(L^2(G_2)\otimes l^2(\mathbb{N}))\otimes B]$.
I should mention, that $\chi$ is homotopy invariant.
It seems like one can use that the compact operators on any Hilbert space with $G$ action is homotopy equivalent to those on $l^2(\mathbb{N})$ with trivial action. This would at least explain the isomorphism. It should be easy since he loses no word in the proof for it. Can somebody help?