Leinert sets and complemented ideals in Fourier algebras

Michael Brannan¹ ; Brian Forrest¹ ; Cameron Zwarich¹

¹

Studia Mathematica, Tome 239 (2017) no. 3, pp. 273-296

Résumé

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.

Détail

DOI : 10.4064/sm8733-3-2017

Keywords: complemented ideals fourier algebra arise naturally class thin sets known leinert sets moreover present explicit example closed ideal mathbb where mathbb group generators complemented mathbb completely complemented establishing appropriate extension result restriction algebras arising leinert sets almost connected group which every complemented ideal completely complemented amenable