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

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