I tried to prove the Green-Julg isomorphism in some notion of KK-theory, but came to this basic problem:
Let $G$ a finite group. Equip $K:=K(\ell^2(G))$ (compact operators) with the left translation induced action $L:G \rightarrow B(\ell^2(G))$ and the right translation induced action $R:G \rightarrow B(\ell^2(G))$. (To be precise, $L = Ad(U)$ of left action $U$ on $\ell^2(G)$.)
Consider two $*$-homomorphisms $$\alpha,\beta:\mathbb{C} \rtimes G \rightarrow K \rtimes_R G,$$ where $\alpha$ is the left regular representation into $K$ and some averaging projection in $G$, and $\beta$ injects canonically into the crossed products and takes some averaging projection in $K$. More precisely, $$\alpha(\lambda \rtimes g)= \lambda L_g \rtimes \frac{1}{|G|} \sum_{h \in G} h$$ and $$\beta(\lambda \rtimes g) = \lambda \sum_{h \in G} \frac{1}{|G|} R_h \rtimes g.$$
Could $\alpha$ and $\beta$ be homotopic?
Or could at least $\alpha(p)$ and $\beta(p)$ be homotopic projections for some projection $p$?
Some remarks:
(1) One could further embed $\alpha$ and $\beta$ into $K \otimes K$, and there it would be easy to exchange the coordinates by a homotopy. But this would be too much loss of information.
(2) $L_g$ and $R_g$ commute. Possibly, the image of $\alpha$ in the $K$-part is the fixed point algebra of $K$ under the $R$-action.
