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.