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

 

Quotients of Fourier algebras, and representations which are not completely bounded


Authors: Yemon Choi and Ebrahim Samei
Journal: Proc. Amer. Math. Soc. 141 (2013), 2379-2388
MSC (2010): Primary 43A30; Secondary 46L07
Published electronically: March 20, 2013
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Abstract: We observe that for a large class of non-amenable groups $ G$, one can find bounded representations of $ {\operatorname {A}}(G)$ on a Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $ {\operatorname {A}}(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras. Partial results are obtained using a modified notion of the Helson set which takes into account operator space structure. In particular, we show that when $ G$ is virtually abelian and $ E$ is a closed subset, the restriction algebra $ {\operatorname {A}} _G(E)$ is completely isomorphic to an operator algebra if and only if $ E$ is finite.


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Additional Information

Yemon Choi
Affiliation: Department of Mathematics and Statistics, McLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada
Email: choi@math.usask.ca

Ebrahim Samei
Affiliation: Department of Mathematics and Statistics, McLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada
Email: samei@math.usask.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11974-X
PII: S 0002-9939(2013)11974-X
Keywords: Fourier algebra, completely bounded representation, operator algebra, operator space, Leinert set, Helson set
Received by editor(s): August 14, 2011
Received by editor(s) in revised form: October 19, 2011
Published electronically: March 20, 2013
Additional Notes: The first author was supported by NSERC Discovery Grant 402153-2011
The second author was supported by NSERC Discovery Grant 366066-2009
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.