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 $Gamma$. This morphism, called $mu_ au$, is defined in $KK$-theory with coefficients in $mathbb{R}$ by means of the action of the projection $[ au ] in KK_{mathbb{R}}^Gamma(mathbb{C},mathbb{C})$ canonically associated to the group trace of $Gamma$. We show that the corresponding $ au$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $mu_ au$ is functorial with respect to the group $Gamma$.