# Kasparov's descent homomorphism for higher KK groups

I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism $$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$ but what still confuses me are higher $KK$ groups defined by $KK_n^G(A, B):=KK^G(A \hat{\otimes} Cl_n)$ where $Cl_n$ is the $n$-th Clifford-Algebra with trivial $G$-action and $\hat{\otimes}$ is the spatial graded tensor product. How do I get a descent homomorphism $$KK_n^G(A,B) \rightarrow KK_n(A \rtimes G,B \rtimes G)?$$

Thanks.

• If the action of $G$ is trivial on $D$ then we have $(A \otimes D) \ltimes G \sim (A \ltimes G) \otimes D$. Hence your morphisms for higher groups follow the one for $KK_0$ – Bleuderk Feb 27 '17 at 12:53