Operator-algebraic superridigity for SL n (ℤ), n≥3

Abstract

For n≥3, let Γ=SL n (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→U(M) be a group homomorphism such that π(Γ)”=M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π|Λ extends to a homomorphism U(L(Λ))→U(M). This answers, in the special case of SL n (ℤ), a question of A. Connes discussed in [Jone00, p. 86]. 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, thus answering a question of E. Kirchberg.