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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.

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  • $\begingroup$ I'll have to think about it to be sure, but usually these sorts of results follow in the compact case from the Peter-Weyl theorem. $\endgroup$
    – Paul Siegel
    Commented Jan 6, 2014 at 20:03
  • $\begingroup$ @Paul: I appreciate :) $\endgroup$
    – m07kl
    Commented Jan 6, 2014 at 23:39

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