Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$

Does there exist a group $G$ so that the kernel of $\pi$ is the unique non-trivial maximal ideal of $C^*(G)$?

Notice that if there is a $G$ with this property, then the trivial representation of $G$ must be weakly contained in the left regular representation of $G$, so $G$ must be amenable. It feels that no such group should exist, but I've had no luck constructing a second maximal ideal in a general $C^*(G).$