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


Authors: Yoann Dabrowski and Adrian Ioana
Journal: Trans. Amer. Math. Soc. 368 (2016), 4525-4560
MSC (2010): Primary 46L36; Secondary 28D15, 37A20
DOI: https://doi.org/10.1090/tran/6470
Published electronically: October 28, 2015
MathSciNet review: 3456153
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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 :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 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 )$.


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Additional Information

Yoann Dabrowski
Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan UMR 5208, 43 Boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email: dabrowski@math.univ-lyon1.fr

Adrian Ioana
Affiliation: Department of Mathematics, University of California, San Diego, California 90095-1555
Email: aioana@ucsd.edu

DOI: https://doi.org/10.1090/tran/6470
Received by editor(s): May 4, 2013
Received by editor(s) in revised form: May 4, 2014
Published electronically: October 28, 2015
Additional Notes: The first author was partially supported by ANR grant NEUMANN
The second author was partially supported by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402, and a Sloan Foundation Fellowship
Article copyright: © Copyright 2015 American Mathematical Society