Weak spectral synthesis in Fourier algebras of coset spaces
Let $G$ be a locally compact group, $K$ a compact subgroup of $G$ and $A(G/K)$ the Fourier algebra of the coset space $G/K$. Applying results from [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008), 987–1002], we establish injection and localization theorems relating weak spectral sets and weak Ditkin sets for $A(G/K)$ to such sets for $A(H/H\cap K)$, where $H$ is a closed subgroup of $G$. We also prove some results towards the analogue of Malliavin's theorem for weak spectral synthesis in $A(G/K)$ and give illustrating examples.