The Baum--Connes conjecture localised at the unit element of a discrete group

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

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