The Fourier algebra of \(SL(2,\mathbb{R})\rtimes\mathbb{R}^ n\), \(n\geq2\), has no multiplier bounded approximate unit
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.
scientific article; zbMATH DE number 537340
scientific article