Fourier algebras on homogeneous spaces
type: Item
instance of: scholarly article
title: Fourier algebras on homogeneous spaces
zbMATH Open document ID: 1219.43004
author: N. Shravan Kumar
author: K. Parthasarathy
published in: Bulletin des Sciences Mathématiques
publication date: 2011-03-17T00:00:00Z
review text: The authors study the spectral synthesis properties of the Fourier algebra \(A(G/K)\) of a homogeneous space \(G/K\), in which \(K\) is a (non-normal) compact subgroup of a locally compact group \(G\). The Fourier algebra of these coset spaces has been introduced by \textit{B. Forrest} [Rocky Mt. J. Math. 28, 173--190 (1998; Zbl 0922.43007)]. In Section 3, the authors prove an injection theorem which generalizes earlier results of \textit{E. Kaniuth} and \textit{A. T. Lau} [Proc. Am. Math. Soc. 129, 3253--3263 (2001; Zbl 0976.43002)] and \textit{H. Reiter} (see [Classical harmonic analysis and locally compact groups. Oxford Univ. Press (1968; Zbl 0165.15601)]). The injection theorem of the authors (Theorem 3.2) is as follows: with \(G\) and \(K\) as above, let \(H\) be a closed subgroup of \(G\) containing \(K\). Then a closed set \(\widetilde E\subset H/K\) is of (weak) synthesis for \(A(G/K)\) if and only if it is of (weak) synthesis of \(A(H/K)\). The special case of \(K= \{e\}\) reduces to Kaniuth and Lau's result referred to above. Theorem 3.12 is of similar nature. Let \(K\setminus G\) be the space of right cosets and let the Varopoulos algebra of \(G\) and \(K\setminus G\) be defined by the Haagerup tensor product \(V(G, K\setminus G)= C(G)\otimes^h C(K\setminus G)\). For a closed subset \(\widetilde E\subset G/K\), let \(\widetilde E^\natural:= \{(s,\overline t)\in G\times K\setminus G: s\cdot(t^{-1} K)\in\widetilde E\}\). In that case, if \(G\) is compact, then \(\widetilde E\) is a set of (weak) synthesis for \(A(G/K)\) if and only if \(\widetilde E^\natural\) is a set of (weak) synthesis for \(V(G, K\setminus G)\). Following closely the work of \textit{N. Spronk} [Proc. Lond. Math. Soc. 89, 161--192 (2004; Zbl 1047.43008)], in Section 4, the authors obtain a complete isometry between completely bounded multipliers of \(A(G/K)\) and the space \(V^\infty_{\text{inv}}(G, K\setminus G)\) (Theorem 4.7). This result is then used in the final section to prove th…
Mathematics Subject Classification ID: 43A45
zbMATH DE Number: 5867241
zbMATH Keywords: Fourier algebra
zbMATH Keywords: Varopoulos algebra
zbMATH Keywords: spectral synthesis
zbMATH Keywords: operator synthesis
zbMATH Keywords: \(w^*\) Haagerup tensor product
zbMATH Keywords: completely bounded multipliers
zbMATH Keywords: Schur multipliers
reviewed by: M. Sangani Monfared
MaRDI profile type: Publication
full work available at URL: j.bulsci.2010.03.003
OpenAlex ID: W2069035921
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DOI: 10.1016/J.BULSCI.2010.03.003
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description: scientific article; zbMATH DE number 5867241