Special subsets of discrete groups and Arens regularity of ideals of Fourier and group algebras

Let $ \mathscr A$ be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual $\mathscr A^{\ast\ast}$. We refer to such an algebra as a WeSeBai algebra.

In the present paper, we examine the Arens regularity properties of closed ideals of algebras in this class and observe that, although WeSeBai algebras themselves are always strongly Arens irregular, a variety of Arens regularity properties can be found among their closed ideals.

After characterizing Arens regular ideals and strongly Arens irregular ideals, we focus on the main examples of WeSeBai algebras: the convolution group algebras $L^1(G)$, $G$ compact, and the Fourier algebras $A(\varGamma )$, $\varGamma $ discrete and amenable. We find examples of Arens regular ideals in $L^1(G)$ and $A(\varGamma )$, reflexive and nonreflexive, and examples of strongly Arens irregular ideals that are not in the WeSeBai class. For this, we introduce a new class of Riesz sets that are not $\varLambda (p)$, for any $p \gt 1$, and show how to obtain them in a broad family of noncommutative groups. As a consequence of our approach, we prove that every infinite Abelian group contains a Rosenthal set that is not $\varLambda (p)$, for any $p \gt 0$.