Operator-algebraic superrigidity for $SL_n(\mathbb Z),n\geq 3$

For $n\geq 3,$ let $\Gamma=SL_n(\mathbb Z).$ We prove the following superridigity result for $\Gamma$ in the context of operator algebras. Let $L(\Gamma)$ be the von Neumann algebra generated by the left regular representation of $\Gamma.$ Let $M$ be a finite factor and let $U(M)$ be its unitary group. Let $\pi: \Gamma\to U(M)$ be a group homomorphism such that $\pi(\Gamma)''=M.$ Then \begin{itemize} \item[(i)] either $M$ is finite dimensional, or \item [(ii)] there exists a subgroup of finite index $\Lambda$ of $\Gamma$ such that $\pi|_\Lambda$ extends to a homomorphism $U(L(\Lambda))\to U(M).$ \end{itemize} The result is deduced from a complete description of the tracial states on the full $C^*$--algebra of $\Gamma.$ As another application, we show that the full $C^*$--algebra of $\Gamma$ has no faithful tracial state.