The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras
type: Item
instance of: scholarly article
title: The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras
zbMATH Open document ID: 1208.43003
author: Michael Brannan
author: Ebrahim Samei
published in: Journal of Functional Analysis
publication date: 2010-09-10T00:00:00Z
review text: The paper under review deals with the similarity problem for the Fourier algebra \({\mathcal A}(G)\) of a locally compact group \(G\). Recall that for a general involutive Banach algebra \({\mathcal A}\) the similarity problem asks if every bounded representation of \({\mathcal A}\) as operators on a Hilbert space \({\mathcal H}\) is similar to a \(\star\)-representation. The similarity problem for \(C^*\)-algebras is known as the Kadison Similarity Problem, which in its full generality is still open. The authors consider the similarity problem for the Fourier algebra \({\mathcal A}(G)\), which may be considered as the dual version of the similarity problem for \(L^1(G)\) from the point of view of locally compact quantum groups. Effros and Ruan used this point of view to define Hopf algebraic structures on \({\mathcal A}(G)\) and asked for a similarity problem in this setting, i.e. whether every completely contractive representation of \({\mathcal A}(G)\) on a Hilbert space is in fact a \(\ast\)-representation. The second question the paper addresses is as follows, whether every completely bounded representation of \({\mathcal A}(G)\) on a Hilbert space is similar to a completely contractive representation. The main results give partial affirmative answers to both of these questions. The authors obtain an analogous characterization for Fourier algebras to that of Haagerup's for \(C^*\)-algebras: the similarity problem for the Fourier algebra \({\mathcal A}(G)\) has a negative answer if and only if there is a bounded representation of \({\mathcal A}(G)\) which is not completely bounded. Furthermore, it is shown that there is a close connection between the similarity problem for \({\mathcal A}(G)\) and the invertibility of corepresentations of \(VN(G)\).
reviewed by: Franz Luef
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 46L07
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 22D20
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 5782194
zbMATH Keywords: Fourier algebra
zbMATH Keywords: group von Neumann algebras
zbMATH Keywords: completely bounded homomorphisms
zbMATH Keywords: corepresentations
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.06.011
OpenAlex ID: W2168070361
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DOI: 10.1016/J.JFA.2010.06.011
description: scientific article; zbMATH DE number 5782194