Let $\Gamma $ be a lattice in a non-compact simple Lie group G. We prove that the canonical map from the full $C^*$-algebra $C^*(\Gamma )$ to the multiplier algebra $M(C^*(G))$ is not injective in general (it is never injective if $G$ has Kazhdan's property $(T)$, and not injective for many lattices either in $SO(n,1)$ or $SU(n, 1)$). For a locally compact group $G$, Fell introduced a property $(WF3)$, stating that for any closed subgroup $H$ of $G$, the canonical map from $C^*(H)$ to $M(C^*(G))$ is injective. We prove that, for an almost connected $G$, property $(WF3)$ is equivalent to amenability.