Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary

Abstract

We denote by $\mathbb{T}$ the unit circle and by $\mathbb{D}$ the unit disc. Let $\mathcal{B}$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$, endowed with the pointwise product. We assume that $\mathcal{B}$ is continously imbedded in the disc algebra and satisfies the following conditions:

The space of polynomials is a dense subset of $\mathcal{B}$.

$\lim_{n\to +\infty}\|z^n\|_{\mathcal{B}}^{1/ n}=1$.

There exist $k \geq 0$ and $C > 0$ such that $$ \bigl| 1- |\lambda| \bigr|^{k} \bigl\| f \bigr\|_{\mathcal{B}} \leq C \bigl\| (z-\lambda) f \bigr\|_{\mathcal{B}}, \quad (f \in \mathcal{B},\, |\lambda| < 2).$$

When $\mathcal{B}$ satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals $I$ of $\mathcal{B}$ with countable hull $h(I)$, where $$h(I) = \bigl\{ z \in \overline{\mathbb{D}} : f(z) = 0, \quad (f \in I) \bigr\}.$$ Then, we apply this result to many algebras for which the structure of all closed ideals is unknown. We consider, in particular, the weighted algebras $\ell^1(\omega$) and $L^1(\mathbb{R}^{+},\omega)$.