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Abstract

Let $G$ be a locally compact amenable group, and $A(G)$ and $B(G)$ the Fourier and Fourier–Stieltjes algebras of $G$. For a closed subset $E$ of $G$, let $J(E)$ and $k(E)$ be the smallest and largest closed ideals of $A(G)$ with hull $E$, respectively. We study sets $E$ for which the ideals $J(E)$ or/and $k(E)$ are $\sigma (A(G),C^{\ast }(G))$-closed in $A(G)$. Moreover, we present, in terms of the uniform topology of $C_{0}(G)$ and the weak$^{\ast }$ topology of $B(G)$, a series of characterizations of sets obeying synthesis. Finally, closely related to the above issues, we present a series of results about closed sets of uniqueness (i.e. closed sets $E$ for which $\overline {J(E)}^{w^{\ast }}=B(G)$).