Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the vector state $\tau$ corresponding to $\delta_e$, that is
$\tau(T) = \langle T\delta_x, \delta_x\rangle$
where $T\in C^*(G)$. The left kernel of $\tau$, that is, $\mathscr{N}=\{T\in C^*(G)\colon \tau(T^*T)=0\}$ is an ideal of $C^*(G)$.
Is there example of a group $G$ as above for which there is a projection $p\in \mathscr{N}$ such that $\mathscr{N} = \mathscr{N}p$?