Unbounded derivations, free dilations and indecomposability results for II$_1$ factors

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 $\Gamma\curvearrowright (X,\mu)$.