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L 2-rigidity in von Neumann algebras

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Abstract

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.

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  1. Department of Mathematics, University of California at Berkeley, 970 Evans Hall, Berkeley, CA, 94720-3840, USA

    Jesse Peterson

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Peterson, J. L 2-rigidity in von Neumann algebras. Invent. math. 175, 417–433 (2009). https://doi.org/10.1007/s00222-008-0154-6

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  • DOI: https://doi.org/10.1007/s00222-008-0154-6

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