The naive Baum–Connes conjecture in KK-theory

Let $G$ be a topological group with a universal proper space $\mathcal E G$. Let $A$ be a $C^*$-algebra and let $B$ be a $G$-algebra. Define the naive topological $KK$-groups by

$$KK^{\mathrm{naive}}_{*}(G;A,B)=\operatorname*{colim}_{\substack{Y\subseteq\mathcal E G\G\text{-compact}}}KK^G_{*}(A(Y),B),\qquad *=0,1,$$

and let $\beta_G^{A,B}:KK^{\mathrm{naive}}_{*}(G;A,B)\to KK_{*}(A,B\rtimes_{\mathrm r}G)$ be the reduced naive assembly map. The naive Baum–Connes conjecture in $KK$-theory. For given $A$ and $B$, $G$ satisfies the naive Baum–Connes conjecture for $(A,B)$ if $\beta_G^{A,B}$ is an isomorphism of abelian groups.

The claim is presented as a naive generalization of the Baum–Connes conjecture. Its status is not resolved by the supplied text; the paper explains that the naive formulation has a defect because its left-hand side is not $\sigma$-additive in the first variable.