L^2-rigidity in von Neumann algebras

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 II_1 factors which are non-prime, have property $\Gamma$, or are weakly rigid. As a consequence we obtain that if $M$ is a free product of diffuse von Neumann algebras, or if $M=L\Gamma$ where $\Gamma$ is a finitely generated group with $b_1^{(2)}(\Gamma )>0$, then any nonamenable regular subfactor of $M$ is prime and does not have properties $\Gamma$ or (T). In particular this gives a new approach for showing primeness of all nonamenable subfactors of a free group factor thus recovering a well known recent result of N. Ozawa.

19 pages

References in corpus (1)

• A 1-cohomology characterization of property (T) in von Neumann algebras

Cited by in corpus (5)

• On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra
• On a class of II_1 factors with at most one Cartan subalgebra II
• Noncommutative Riesz transforms -- a probabilistic approach
• The symbiosis of C*- and W*-algebras
• On the Superrigidity of Malleable Actions with Spectral Gap