L^2-rigidity in von Neumann algebras

TL;DR

This paper introduces L^2-rigidity for von Neumann algebras, a property extending property (T), and demonstrates its implications for the structure and primeness of subfactors in various nonamenable factors.

Contribution

It defines L^2-rigidity as a new property for von Neumann algebras and explores its inheritance and implications for primeness and properties of subfactors.

Findings

L^2-rigidity passes to normalizers.

Nonamenable II_1 factors with certain properties are L^2-rigid.

Nonamenable subfactors of free product factors are prime and lack property (T) or $ extGamma$.

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.