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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rigid actions need not be strongly ergodic


Authors: Adrian Ioana and Stefaan Vaes
Journal: Proc. Amer. Math. Soc. 140 (2012), 3283-3288
MSC (2010): Primary 28D15, 46L10; Secondary 37A20
Published electronically: February 3, 2012
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Abstract: A probability measure preserving action $ \Gamma \curvearrowright (X,\mu )$ is called rigid if the inclusion of $ \mathrm {L}^\infty (X)$ into the crossed product $ \mathrm {L}^\infty (X) \rtimes \Gamma $ has the relative property (T) in the sense of Popa. We give examples of rigid, free, probability measure preserving actions that are ergodic but not strongly ergodic. The same examples show that rigid actions may admit non-rigid quotients.


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Additional Information

Adrian Ioana
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Address at time of publication: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: adiioana@math.ucla.edu, aioana@math.ucsd.edu

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/S0002-9939-2012-11261-4
PII: S 0002-9939(2012)11261-4
Received by editor(s): September 21, 2010
Received by editor(s) in revised form: April 4, 2011
Published electronically: February 3, 2012
Additional Notes: The first author was supported by a Clay Research Fellowship
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.
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.