  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.
