Stable Arens regularity of pointwise-multiplication ℓ₁(ℤ)

Determine whether the Banach algebra ℓ₁(ℤ) equipped with pointwise multiplication has stable Arens regularity, meaning that all sufficiently small perturbations of its multiplication remain Arens regular.

Background

The paper defines stable Arens regularity (SAR) for an Arens-regular Banach algebra as the existence of a positive threshold K such that every perturbation of the multiplication with size at most K remains Arens regular. It establishes SAR for Banach function algebras equivalent to uniform algebras and notes that every C*-algebra has this property.

The authors then ask specifically whether ℓ₁(ℤ), with pointwise multiplication, also possesses SAR. This is left unresolved and is presented as a natural question following the paper’s positive stability results for other classes of Banach algebras.