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 $\delta :M\to L^2(M)\bar{\otimes }{L}^2(M)$ 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$\delta^*(1\otimes 1)\in M_0$),thenweshowthat$M$ has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups $\mathrm{\Gamma }$, defined through the existence of an unbounded cocycle $b:\Gamma\rightarrow \mathbb C(\Gamma/\Lambda)$,forsomesubgroup$\Lambda<\Gamma$,suchthattheII$_1$factor$L^{\infty}(X)\rtimes\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $\mathrm{\Gamma }\curvearrowright (X,\mu )$.