Unbounded derivations, free dilations and indecomposability results for II_1 factors
Résumé
We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II_1 factor M is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation \delta:M\rightarrow L^2(M)\bar{\otimes}L^2(M) whose domain contains a non-amenability set, then M is prime. If \delta is moreover "algebraic" (i.e. its domain M_0 is finitely generated, \delta(M_0)\subset M_0\otimes M_0 and \delta^*(1\otimes 1)\in M_0), then we show that M has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups \Gamma, defined through the existence of an unbounded cocycle b:\Gamma\rightarrow \mathbb C(\Gamma/\Lambda), for some subgroup \Lambda<\Gamma, such that the II_1 factor L^{\infty}(X)\rtimes\Gamma has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action \Gamma\curvearrowright (X,\mu).