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Let $ G $ be a second countable locally compact group.

Let $ A $ and $ B $ be two $G$-$C^*$-algebras.

Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $.

Could you tell me please, when $ KK^G (A, B) $ is a free $ \mathrm{Rep}(G) $-module of finite rank ?

$ \mathrm{Rep}(G) $ is the representation ring of $ G $.

Thanks in advance for your help.

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  • $\begingroup$ The question seems rather unmotivated - what are the motivating examples or theorems? Are there known cases in the literature where $KK^G$ does have the properties you are asking for? $\endgroup$
    – Yemon Choi
    Sep 14, 2021 at 23:42

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