## Tohoku Mathematical Journal

### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 59, Number 3 (2007), 441-454.

Dates
First available in Project Euclid: 11 October 2007

https://projecteuclid.org/euclid.tmj/1192117987

Digital Object Identifier
doi:10.2748/tmj/1192117987

Mathematical Reviews number (MathSciNet)
MR2365350

Zentralblatt MATH identifier
1152.43004

#### Citation

Parthasarathy, Krishnan; Prakash, Rajendran. Spectral synthesis in the Fourier algebra and the Varopoulos algebra. Tohoku Math. J. (2) 59 (2007), no. 3, 441--454. doi:10.2748/tmj/1192117987. https://projecteuclid.org/euclid.tmj/1192117987

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