Spectral synthesis in the Fourier algebra and the Varopoulos algebra

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.