Operator-algebraic superrigidity for $S{L}_n(\mathbb{Z}),n\ge 3$

Abstract

For $n\ge 3,$ let $\Gamma =S{L}_n(\mathbb{Z}).$ 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 $\pi (\Gamma {)}^{\prime \prime }=M.$ 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 $C^{\ast }$--algebra of $\Gamma .$ As another application, we show that the full $C^{\ast }$--algebra of $\Gamma$ has no faithful tracial state.

30 pages; typos corrected

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