  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.
