The Green-Julg Theorem

I am currently trying to understand the general Green-Julg theorem, where $G$ is a compact group, $A$ and $B$ are $G$-$C^*$-algebras, and where $G$ acts trivially on $A$. The Green-Julg theorem states that there is an isomorphism $$\mathrm{KK}^G(A,B) \rightarrow \mathrm{KK}(A, B\rtimes G).$$ Unfortunately, in all of the papers I can find, it's always the special case $A=\mathbb{C}$. Does anybody know a good paper where a proof is given for the general case?

Thank you.

• If true what you say, I think you can simply copy the usual Green-Julg proof and inherit the additional $\varphi:A\rightarrow L(E)$ of the Kasparov cycle $(\varphi,E)\in KK^G(A,B)$. Because $G$ acts trivially, $A \rtimes G = A \otimes G$ under the descent in the proof. Mar 10, 2018 at 12:05

For $$G$$ a finite group, this is part 3 of Theorem 1.22 in the preprint A stable ∞-category for equivariant KK-theory by Ulrich Bunke, Alexander Engel and Markus Land.
More precisely, they prove an adjunction $$\operatorname{Res}_G: \operatorname{KK} \rightleftarrows \operatorname{KK}^G: - \rtimes G$$ of $$\infty$$-categories, where $$\operatorname{Res}_G: \operatorname{KK} \to \operatorname{KK}^G$$ is induced by the functor $$C^*{\rm Alg} \to GC^*{\rm Alg}$$ that equips a $$C^*$$-algebra $$A$$ with the trivial $$G$$-action. This induces an adjunction on homotopy categories. If $$A$$ is a $$C^*$$-algebra and $$B$$ is a $$G$$-$$C^*$$-algebra, this gives desired isomorphism \begin{aligned} \mathrm{KK}^G(\operatorname{Res}_GA,B) \cong\mathrm{KK}(A, B\rtimes G). \end{aligned}