Differential and maximal ideals of the ultrametric Corona algebra

Abstract

International audienceLet $K$ 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. Maximal ideals of infinite codimension are examined in connection with ultrafilters on $D$. Four classes of ultrafilters on $D$ are considered, defining a null ideal, or a maximal ideal or an unidentified ideal. A function $f\in A$ tends to $0$ along a sequence of disks $|x-a_n