$\xi \in {\mathrm{KK}}^G(A,B)$ is a ${\mathrm{KK}}^G$-equivalence if and only if $G \ltimes \xi \in KK(G \ltimes A, G \ltimes B)$ is a ${\mathrm{KK}}$-equivalence.

\[thm:KKHod\] Let $G$ be a compact Lie group with Hodgkin condition. Then, for any $G$-algebra $A$ with continuous Rokhlin property, there is a ${\mathrm{KK}}^G$-equivalence from $A$ to $C(G) \otimes A^\alpha$ mapping $[1_A] \in {\mathrm{K}}^G_0(A)$ to $[1_{A^\alpha \otimes C(G)}]$.

By Proposition \[prp:BC\], it suffices to show that $G \ltimes \xi$ is a ${\mathrm{KK}}$-equivalence for $$\xi:=[\psi |_{A^\alpha \otimes C(G)}] \in {\mathrm{KK}}^G(A^\alpha \otimes C(G), A).$$

Let $p$ be the projection in $C^*_\lambda(G)$ corresponding to the trivial representation and consider the canonical inclusions $\iota_1: A^\alpha \to A^\alpha \otimes C(G)$, $\iota_2 : A^\alpha \to A$ and $\mu: {\mathbb C}p \otimes A^\alpha \to C^*_\lambda(G) \otimes A^\alpha$. Since the restriction of $\psi$ to $A^\alpha $ is the identity, we have $\xi \circ [\iota_1]=[\iota_2]$ and hence $(G \ltimes \xi) \circ (G \ltimes [\iota_1])=G \ltimes [\iota_2]$. We also have $G \ltimes [\iota_k] \circ [\mu]$ are ${\mathrm{KK}}$-equivalences for $k=1,2$ since the isomorphism in Lemma \[lem:avbdl\] maps $p$ to a $1$-dimensional projection in ${\mathbb{K}}$. Hence $G \ltimes \xi=(G \ltimes [i_2] \circ [\mu])(G \ltimes [i_1] \circ [\mu])^{-1}$ is a ${\mathrm{KK}}$-equivalence.

\[rmk:inner\] In fact, ${\mathrm{KK}}^G$-equivalence $A \sim B \otimes C(G)$ 