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- W2963479330 abstract "We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain regularity 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 $delta :Mrightarrow L^2(M)bar {otimes }L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $delta$ is moreover âalgebraicâ (i.e. its domain $M_0$ is finitely generated, $delta (M_0)subset M_0otimes M_0$ and $delta ^*(1otimes 1)in M_0$), 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 certain subgroups $Lambda <Gamma$, 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 $Gamma curvearrowright (X,mu )$." @default.
- W2963479330 created "2019-07-30" @default.
- W2963479330 creator A5052788604 @default.
- W2963479330 creator A5070665706 @default.
- W2963479330 date "2015-10-28" @default.
- W2963479330 modified "2023-10-13" @default.
- W2963479330 title "Unbounded derivations, free dilations, and indecomposability results for II$_1$ factors" @default.
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- W2963479330 doi "https://doi.org/10.1090/tran/6470" @default.
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