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


Yemon Choi*, Ebrahim Samei

* Corresponding author


2010 MSC: 43A30 (primary); 46L07 (secondary)


Appeared as Proc. Amer. Math. Soc. 141 (2013), no. 7, 2379--2388.

Preprint version available at arXiv 1104.2953


[ Math Review | Zentralblatt ]


We observe that for a large class of non-amenable groups G, one can find bounded representations of A(G) on a Hilbert space which are not completely bounded. We also consider restriction algebras obtained from 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 AG(E) is completely isomorphic to an operator algebra if and only if E is finite.


This paper has its genesis in a conversation the two authors had while getting coffee after teaching.

The main construction from the first part of this paper has been extended to the quantum group setting, in work of the second author together with M. Brannan and M. Daws: see Münster J. Math. 6 (2013), 445–482.

Comparison of the final submitted version with the final published version will show that at least one of the authors is overly fond of semicolons, while AMS copyeditors are not. It will also show that once again, copy-editors seem to inist on prepending the definite article to named concepts (in this case "Helson set", on a different occasion "Hochschild cohomology") where this doesn't seem quite right. After all, would one talk about "a weaker notion of the thin set"?

(Both on this occasion and on the one with "Hochschild cohomology", I remember being in a distracted rush while trying to get the page proofs read and returned on time. Had I spotted these additions of the definite article, I would have tried to get them changed. Oh well, there's always the next time...)

Yemon Choi