Tohoku Mathematical Journal
2007
September
SECOND SERIES VOL. 59, NO. 3

Tohoku Math. J.
59 (2007), 441-454

Title SPECTRAL SYNTHESIS IN THE FOURIER ALGEBRA AND THE VAROPOULOS ALGEBRA

Author Krishnan Parthasarathy and Rajendran Prakash

(Received March 13, 2006, revised January 17, 2007)
Abstract. The objects of study in this paper are sets of spectral synthesis for the Fourier algebra $A(G)$ of a locally compact group and the Varopoulos algebra $V(G)$ of a compact group with respect to submodules of the dual space. Such sets of synthesis are characterized in terms of certain closed ideals. For a closed set in a closed subgroup $H$ of $G$, the relations between these ideals in the Fourier algebras of $G$ and $H$ are obtained. The injection theorem for such sets of synthesis is then a consequence. For the Fourier algebra of the quotient modulo a compact subgroup, an inverse projection theorem is proved. For a compact group, a correspondence between submodules of the dual spaces of $A(G)$ and $V(G)$ is set up and this leads to a relation between the corresponding sets of synthesis.

2000 Mathematics Subject Classification. Primary 43A45; Secondary 43A77, 43A85.
Key words and phrases. Fourier algebra, Varopoulos algebra, spectral synthesis.

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