The Fourier algebra of \(SL(2,\mathbb{R})\rtimes\mathbb{R}^ n\), \(n\geq2\), has no multiplier bounded approximate unit
type: Item
instance of: scholarly article
title: The Fourier algebra of \(SL(2,\mathbb{R})\rtimes\mathbb{R}^ n\), \(n\geq2\), has no multiplier bounded approximate unit
zbMATH Open document ID: 0791.43004
DOI: 10.1007/BF01459526
author: Brian Dorofaeff
published in: Mathematische Annalen
publication date: 1994-06-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165155
review text: For each fixed positive integer \(n>1\) we denote by \(G^{(n)}\) the semidirect product Lie group \(SL(2,{\mathcal R}) \rtimes {\mathcal R}^ n\), where \(SL(2,{\mathcal R})\) is the special linear group of \(2 \times 2\) real matrices of determinant one acting on \({\mathcal R}^ n\) by the unique irreducible representation of dimension \(n\). We show that the Fourier algebra \(A(G^{(n)})\) of \(G^{(n)}\), for each \(n=2,3,\dots,\) fails to admit an approximate unit of functions from the Fourier algebra which has the additional property of being uniformly bounded in the multiplier norm. This generalises a result of U. Haagerup (1986), who has shown that \(SL(2,{\mathcal R}) \rtimes {\mathcal R}^ 2\) does not admit an approximate unit of this type on its Fourier algebra. Hence for each group \(G^{(n)}\) the Haagerup invariant \(\Lambda(G^{(n)})\) takes the extended integer value infinity.
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22D25
zbMATH DE Number: 537340
zbMATH Keywords: semidirect product
zbMATH Keywords: special linear group
zbMATH Keywords: irreducible representation
zbMATH Keywords: Fourier algebra
zbMATH Keywords: approximate unit
zbMATH Keywords: uniformly bounded
zbMATH Keywords: Haagerup invariant
MaRDI profile type: Publication
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OpenAlex ID: W2051567233
description: scientific article; zbMATH DE number 537340