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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

HNN extensions and unique group measure space decomposition of II$ _1$ factors


Authors: Pierre Fima and Stefaan Vaes
Journal: Trans. Amer. Math. Soc. 364 (2012), 2601-2617
MSC (2010): Primary 46L36; Secondary 20E06, 28D15, 46L54
Published electronically: January 6, 2012
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Abstract: We prove that for a fairly large family of HNN extensions $ \Gamma $, the group measure space II$ _1$ factor $ \textup {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 $ \textup {L}^\infty (X) \rtimes \Gamma $ entirely remembers the group action from which it was constructed.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05415-8
PII: S 0002-9947(2012)05415-8
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.