Arens regularity of ideals in $A(G)$, $A_{cb}(G)$ and $A_M(G)$

TL;DR

This paper investigates when ideals in Fourier algebras and their variants are Arens regular, establishing that non-zero ideals are Arens regular only for discrete groups and finite-dimensional ideals with bounded approximate identities.

Contribution

It characterizes Arens regular ideals in Fourier algebras and their closures, linking regularity to discreteness of the underlying group and finite-dimensionality.

Findings

Non-zero ideals are Arens regular only if the group is discrete.

Ideals with bounded approximate identities are Arens regular iff finite-dimensional.

Provides conditions for Arens regularity in Fourier algebra ideals.

Abstract

In this paper, we look at the question of when various ideals in the Fourier algebra $A(G)$ or its closures $A_M(G)$ and $A_{cb}(G)$ in, respectively, its multiplier and $cb$-multiplier algebra are Arens regular. We show that in each case, if a non-zero ideal is Arens regular, then the underlying group $G$ must be discrete. In addition, we show that if an ideal $I$ in $A(G)$ has a bounded approximate identity, then it is Arens regular if and only if it is finite-dimensional.