HNN extensions and unique group measure space decomposition of II$_1$ factors
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- by Pierre Fima and Stefaan Vaes PDF
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Abstract:
We prove that for a fairly large family of HNN extensions $\Gamma$, the group measure space II$_1$ factor $\mathrm {L}^\infty (X) \rtimes \Gamma$ given by an arbitrary free ergodic probability measure preserving action of $\Gamma$ has a unique group measure space Cartan subalgebra up to unitary conjugacy. From this we deduce new examples of W$^*$-superrigid group actions, i.e. where the II$_1$ factor $\mathrm {L}^\infty (X) \rtimes \Gamma$ entirely remembers the group action from which it was constructed.References
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Additional Information
- Pierre Fima
- Affiliation: Department of Mathematics, K. U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium
- Address at time of publication: Institute of Mathematics of Jussieu, University Denis Diderot Paris 7, 175, rue du Chevaleret, F-75013 Paris, France
- Email: pierre.fima@wis.kuleuven.be, pfima@math.jussieu.fr
- Stefaan Vaes
- Affiliation: Department of Mathematics, K. U. Leuven, Celestijnenlaan 200B, B–3001 Leuven, Belgium
- Email: stefaan.vaes@wis.kuleuven.be
- Received by editor(s): June 15, 2010
- Received by editor(s) in revised form: July 6, 2010
- Published electronically: January 6, 2012
- Additional Notes: The first author was supported by ERC Starting Grant VNALG-200749.
The second author was partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0231.07 of the Research Foundation, Flanders (FWO) and K. U. Leuven BOF research grant OT/08/032. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2601-2617
- MSC (2010): Primary 46L36; Secondary 20E06, 28D15, 46L54
- DOI: https://doi.org/10.1090/S0002-9947-2012-05415-8
- MathSciNet review: 2888221