Fourier algebras on homogeneous spaces (Q630622)

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 the following result: if \(G\) is second countable locally compact, then a closed subset \(\widetilde E\subset G/K\) is a set of local spectral synthesis for \(A(G/K)\) if and only if \(\widetilde E^\natural\) is a set of operator synthesis with respect to \(m_{K\setminus G}\times m_G\), where \(\widetilde E^\natural= \{(\overline t,s)\in K\setminus G\times G: s\cdot (t^{-1}K)\in\widetilde E\}\) (cf. Theorems 5.4 and 5.9.). This theorem extends to the homogeneous spaces earlier results by Ludwig, Turowska, Shulman, and Spronk.