Leinert sets and complemented ideals in Fourier algebras

TL;DR

This paper explores the structure of complemented ideals in Fourier algebras of locally compact groups, linking them to Leinert sets and establishing conditions for their complete complementarity, with implications for group amenability.

Contribution

It introduces a natural connection between complemented ideals and Leinert sets in Fourier algebras and provides an explicit example in free groups, also characterizing amenability through ideal complementarity.

Findings

Complemented ideals in $A(G)$ relate to Leinert sets.

An explicit example of a non-completely complemented ideal in $A( ext{free group})$ is provided.

Groups where all complemented ideals are completely complemented are shown to be amenable.

Abstract

Let $G$ be a locally compact group. 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 also present an explicit example of a closed ideal in $A(\mathbb{F}_{N})$, 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.