Inventiones mathematicae

, Volume 175, Issue 2, pp 417-433

First online:

L 2-rigidity in von Neumann algebras

  • Jesse PetersonAffiliated withDepartment of Mathematics, University of California at Berkeley Email author 

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We introduce the notion of L 2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L 2-rigidity passes to normalizers and is satisfied by nonamenable II1 factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M=LΓ where Γ is a finitely generated group with β1 (2)(Γ)>0, then any nonamenable regular subfactor of M is prime and does not have properties Γ or (T). In particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of N. Ozawa.