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Abstract

We show how complemented ideals in the Fourier algebra $A(G)$ of $G$ arise naturally from a class of thin sets known as Leinert sets. Moreover, we present an explicit example of a closed ideal in $A(\mathbb {F}_{N})$, where $\mathbb {F}_{N}$ is the free group on $N \ge 2$ generators, that is complemented in $A(\mathbb {F}_{N})$ but it is not completely complemented. Then by establishing an appropriate extension result for restriction algebras arising from Leinert sets, we show that any almost connected group $G$ for which every complemented ideal in $A(G)$ is also completely complemented must be amenable.