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Quotients of Fourier algebras, and representations which are not completely bounded. (English) Zbl 1275.43005
The Fourier algebra \(A(G)\) of a locally compact group inherits a canonical operator space structure as the unique isometric predual of the group von Neumann algebra of \(G\). In a recent paper [J. Funct. Anal. 259, No. 8, 2073–2097 (2010; Zbl 1208.43003)], M. Brannan and E. Samei proved that every completely bounded homomorphism, \(\phi : A(G) \rightarrow B(H)\), from the Fourier algebra into the bounded linear maps on a Hilbert space is similar to a \(*\)-representation when \(G\) is a [SIN]-group.
In this paper, the authors show that this result cannot be extended to arbitrary bounded homomorphisms. The proof of this fact is detailed and clean. In the second part of the paper, the authors consider restriction algebras of \(A(G)\). Concretely, if \(E\) is a closed subset of a locally compact group \(G\), denote \(I_E = \{ f \in A(G) : f_{| E}=0\}\) and consider \(A_G(E)= A(G)/I_E\). The paper deals with the question: are there infinite sets \(E\) for which \(A_G(E)\) is completely bounded isomorphic to an operator algebra? Using an analogue of Helson sets, the authors answer the question in the negative in some cases.

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
46L07 Operator spaces and completely bounded maps
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