Survey on the Kakutani problem in p-adic analysis I

Abstract

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 pointwise 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 $Mult_1(A,\Vert \ . \ \Vert)$ 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$. By analogy with the Archimedean context, one usually calls {\it ultrametric Corona problem, or ultrametric Kakutani problem} the question whether $Mult_1(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$. In order to recall the study of this problem that was made in several successive steps, here we first recall how to characterize the various continuous multiplicative semi-norms of $A$, with particularly the nice construction of certain multiplicative semi-norms of $A$ whose kernell is neither a null ideal nor a maximal ideal, due to J. Araujo. Here we prove that multbijectivity implies density. The problem of multbijectivity will be described in a further paper.