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Operator-algebraic superrigidity for $SL_n(mathbb Z),ngeq 3$ | | Bachir Bekka
; | | Date: |
4 Sep 2006 | | Subject: | Operator Algebras; Group Theory | | Abstract: | For $ngeq 3,$ 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 factor and let $U(M)$ be its unitary group. Let $pi: Gamma o U(M)$ be a group homomorphism such that $pi(Gamma)’’=M.$ Then
egin{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 $U(L(Lambda)) o U(M).$
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. | | Source: | arXiv, math/0609102 | | Services: | Forum | Review | PDF | Favorites | |
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