Chabert, Echterhoff and Oyono-Oyono proved in [Shapiro's Lemma for topological K-theory of groups] that $K^{top}_*(X\rtimes G;A)\cong K^{top}_*(G;A)$ for any $X\rtimes G$-algebra $A$. They claimed that the result is fairly easy in case $X$ is compact. But I can not see the reason. Is there anybody can help me?

Article [Shapiro's Lemma for topological K-theory of groups] http://download.springer.com/static/pdf/364/art%253A10.1007%252Fs000140300009.pdf?auth66=1389187212_001a3bd0260f8e2cfa7ec9df5d208215&ext=.pdf I think we have to show the following:

$RKK^{G}(X,C_0(X\times T),B)=KK^G(C_0(T),B)$, where $T\subseteq \underline{E}G$ is $G$-compact, and $B$ is $X\rtimes G$-algebra.

Chabert, Echterhoff and Oyono-Oyono proved in [Shapiro's Lemma for topological K-theory of groups] that $K^{top}_*(X\rtimes G;A)\cong K^{top}_*(G;A)$ for any $X\rtimes G$-algebra $A$. They claimed that the result is fairly easy in case $X$ is compact. But I can not see the reason. Is there anybody can help me?

Article [Shapiro's Lemma for topological K-theory of groups] can be found here http://download.springer.com/static/pdf/364/art%253A10.1007%252Fs000140300009.pdf?auth66=1389187212_001a3bd0260f8e2cfa7ec9df5d208215&ext=.pdf

I think we have to show the following to begin with:

$RKK^{G}(X,C_0(X\times T),B)=KK^G(C_0(T),B)$, where $T\subseteq \underline{E}G$ is $G$-compact, and $B$ is $X\rtimes G$-algebra.