Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Abstract

Let I be any index set. We consider the Banach algebra \({\mathbb {C} e+ \ell^2(I)}\) with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For \({I=\mathbb {N}}\) we also characterize all prime z-ideals in this Banach algebra.