%0 Journal Article %T Arens regularity of ideals in $A(G)$, $A_{cb}(G)$ and $A_M(G)$. %J Journal of the Iranian Mathematical Society %I Iranian Mathematical Society %Z 2717-1612 %A Forrest, B. %A Sawatzky, J. %A Thamizhazhagan, A. %D 2023 %\ 01/01/2023 %V 4 %N 1 %P 5-25 %! Arens regularity of ideals in $A(G)$, $A_{cb}(G)$ and $A_M(G)$. %K Fourier algebra %K multipliers %K Arens regularity %K uniformly continuous functionals %K topologically invariant means %R 10.30504/jims.2023.385500.1091 %X 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. %U https://jims.ims.ir/article_171452_757f30a2988d0c0f2d7430947e4be2f2.pdf