Ultrametric Corona Problem and Spherically complete fields
Résumé
Let $K$ be a complete ultrametric algebraically closed field and let $A$ be the Banach $K$-algebra of bounded analytic functions in the ''open'' unit disk $D$ of $K$ provided with the Gauss norm. Let $Mult(A,\Vert \ . \ \Vert)$ be the set of continuous multiplicative semi-norms of $A$ provided with the topology of simple convergence, let $Mult_m(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult(A,\Vert \ . \ \Vert)$ whose kernel is a maximal ideal and let $Mult_a(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult(A,\Vert \ . \ \Vert)$ whose kernel is a maximal ideal of the form $(x-a)A$ with $ a\in D$. We complete the characterization of continuous multiplicative norms of $A$ by proving that the Gauss norm defined on polynomials has a unique continuation to $A$ as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona problem on $A$ lies in two questions: is $Mult_a(A,\Vert \ . \ \Vert)$ dense in $Mult_m(A,\Vert \ . \ \Vert)$? Is it dense in $Mult(A,\Vert \ . \ \Vert)$? In a previous paper, Mainetti and Escassut showed if each maximal ideal of $A$ is the kernel of a unique $\phi \in Mult_m(A,\Vert \ . \ \Vert)$, then the answer to the first question is yes. Particularly, the authors showed that when $K$ is strongly valued, each maximal ideal of $A$ is the kernel of a unique $\phi \in Mult_m(A,\Vert \ . \ \Vert)$. Here we prove that this uniqueness also holds when $K$ is spherically complete, and therefore so does the density of $Mult_a(A,\Vert \ . \ \Vert)$ in $Mult_m(A,\Vert \ . \ \Vert)$.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...