Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary.
Language
en
Article de revue
This item was published in
Publicacions Matemàtiques. 2008, vol. 52, p. 19-56
English Abstract
We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc. Let $\calb$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\bbd$ and continuous on $\overline{\bbd}$, endowed with the ...Read more >
We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc. Let $\calb$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\bbd$ and continuous on $\overline{\bbd}$, endowed with the pointwise product. We assume that $\calb$ is continously imbedded in the disc algebra and satisfies the following conditions: \\ (H1) The space of polynomials is a dense subset of $\calb$. \\ (H2) $\lim_{n\to +\infty}\|z^n\|_{\calb}^{1/ n}=1$.\\ (H3) There exist $k \geq 0$ and $C > 0$ such that \begin{eqnarray*} \big| 1- |\lambda| \big|^{k} \big\| f \big\|_{\calb} \leq C \big\| (z-\lambda) f \big\|_{\calb}, \quad (f \in \calb, |\lambda| < 2) \end{eqnarray*} When $\calb$ satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals $I$ of $\calb$ with countable hull $h(I)$, where $$ h(I) = \big\{ z \in \overline{\bbd} : \, f(z) = 0, \quad (f \in I) \big\}. $$ 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 $L^1(\bbr^{+},\omega)$.Read less <
English Keywords
Ditkin Condition
Closed ideals
Banach algebras
Ditkin Condition.
Origin
Hal imported