The full group C*-algebra of the modular group is primitive

We show that the full group C${}^{\ast }$-algebra of $PSL(n,\mathbb{Z})$ is primitive when $n=2$, and not primitive when $n\ge 3$. Moreover, we show that there exists an uncountable family of pairwise inequivalent, faithful irreducible representations of $C^{\ast }(PSL(2,\mathbb{Z}))$.