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The Fourier algebra of \(SL(2,\mathbb{R})\rtimes\mathbb{R}^ n\), \(n\geq2\), has no multiplier bounded approximate unit. (English) Zbl 0791.43004

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.
Reviewer: B.Dorofaeff

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46J10 Banach algebras of continuous functions, function algebras
43A80 Analysis on other specific Lie groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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References:

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