Operator-algebraic superridigity for SLn(ℤ), n≥3
By Bachir Bekka. 2006
67 citations recorded by the source corpus.
For n≥3, let Γ=SLn(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 SLn(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.
Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1d11982f8eb2529891d6aac36d2e502bdcf8dffa
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In Algebra and Number Theory
- Nathanial P. Brown Narutaka Ozawa
- Gilles Pisier
- Adrian Ioana
- W. Winter
- Thomas Sinclair
- Sorin Popa
- Andreas Thom
- S. Vaes
In Applied Mathematics
In Mathematical Physics
- Nathanial P. Brown Narutaka Ozawa
- Gilles Pisier
- Adrian Ioana
- Sorin Popa
- E. Guentner
- S. Vaes
- Thomas Sinclair
- W. Winter
Citations in the corpus
- About the Connes Embedding Conjecture---Algebraic approaches--- (2012, background): “rem is proved in [KS] for the algebra in Example 3 and in [JP] for the free group algebras, but the proof equally works in the general setting. We note that for some groups Γ, notably for Γ = SL3(Z) ([Be2]), it is possible to describe all the tracial states on k [Γ] . Theorem 6 ([KS]). Let A be a se”