Revistes Catalanes amb Accés Obert (RACO)

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

C. Agrafeuil, M. Zarrabi

Resum


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:
\begin{enumerate}
\item[(H1)] The space of polynomials is a dense subset of $\mathcal{B}$.
\item[(H2)] $\lim_{n\to +\infty}\|z^n\|_{\mathcal{B}}^{1/ n}=1$.
\item[(H3)] There exist $k \geq 0$ and $C > 0$ such that
$$
\bigl| 1- |\lambda| \bigr|^{k} \bigl\| f \bigr\|_{\mathcal{B}} \leq C \bigl\| (z-\lambda) f \bigr\|_{\mathcal{B}}, \quad (f \in \mathcal{B},\, |\lambda| < 2).
$$
\end{enumerate}
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 $L^1(\mathbb{R}^{+},\omega)$.

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