Abstract We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.

中文翻译：

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Baum-Connes 猜想定位于离散群的单位元

摘要 我们构建了一个 Baum-Connes 装配图，定位在离散群 $\Gamma$ 的单位元素上。这种态射称为 $\mu _\tau$，在 $KK$-theory 中定义，系数在 $\mathbb {R}$ 中，通过幂等 $[\tau ]\in KK_{\mathbin { {\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ 典型地关联到 $\Gamma$ 的群迹。我们表明相应的 $\tau$-Baum-Connes 猜想比经典版本弱，但仍然暗示了强 Novikov 猜想。$\mu _\tau$ 的右侧是关于群 $\Gamma$ 的函子。