Survey on the Kakutani problem in p-adic analysis II

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 ultrametric Corona problem, or ultrametric Kakutani problem the question whether $Mult_1(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$. In a previous paper, we have recalled the characterization of a large set of continuous multiplicative semi-norms and why the multbijectivity of the algebra $A$ would solve the Corona problem. Here we prove that multbijectivity in the general case. This implies that $Mult_1(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$, beginning by the case when $\K$ is spherically complete and generalizing next.