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 $δ:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $δ$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $δ(M_0)\subset M_0\otimes M_0$ and $δ^*(1\otimes 1)\in M_0$), 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 $Γ$, defined through the existence of an unbounded cocycle $b:Γ\rightarrow \mathbb C(Γ/Λ)$, for some subgroup $Λ<Γ$, such that the II$_1$ factor $L^{\infty}(X)\rtimesΓ$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $Γ\curvearrowright (X,μ)$.

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