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I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct topological groups, or two distinct locally compact groups, and $ A $ and $B$ ( resp. $ C $ and $ D $ ) are $ G_1 $ - $ C^* $ - algebras ( resp. $ G_2 $ - $ C^* $ - algebras ) ? How to define it precisely in a more general context?

Thanks in advance for your help.

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Isn't this treated in Claude Schochet's 1992 paper "On Equivariant Kasparov Theory and Spanier-Whitehead Duality"? Also, if you want more references, I'd recommend using Google Scholar to see who cites this paper. I note that this led me to a 2011 paper by Uuye about restriction maps (like your situation when $G_2$ is a subgroup of $G_1$).

EDIT: In the light of day, I remembered a better reference. The book K-Theory for Operator Algebras by Bruce Blackadar answers this question in Section 20.5, which begins with the line "We now consider to what extent $KK_G$ is functorial in $G$." This book seems to be the canonical reference for equivariant $KK$-theory.

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  • $\begingroup$ Thank you. But, unfortunately not, this is not treated in the two links that you mentionned. Can you help me to find a book which talk widely about this subject ? Thank you. $\endgroup$
    – YoYo
    Commented Aug 25, 2020 at 21:59
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    $\begingroup$ The question doesn't seem to be well-defined. You are just asking how to construct a map between two very abstract things that have nothing in common ($KK_{G_1}(A,B)$ and $KK_{G_2}(C,D)$). Without more assumptions, the only natural map is the zero map. The answer given here addresses the very natural question of "what happens if there is some natural relationship between $G_1$ and $G_2$ which induces a relationship between $A$ and $C$, and $B$ and $D$?". If you want a different answer, you need to be much more specific about what your question is. $\endgroup$
    – Jamie Gabe
    Commented Aug 27, 2020 at 14:53

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