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

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