Survey on the Kakutani problem in p-adic analysis I - Escassut, Alain (Sarajevo Journal of Mathematics, 2020)
Let be a complete ultrametric algebraically closed field and let A be the Banach -algebra of bounded analytic functions in the ''open'' unit disk D of provided with the Gauss norm. Let Mult(A, . ) be the set of continuous multiplicative semi-norms of A provided with the topology of pointwise convergence, let Mult_m(A, . ) be the subset of the φ Mult(A, . ) whose kernel is a maximal ideal and let Mult_1(A, . ) be the subset of the φ Mult(A, . ) whose kernel is a maximal ideal of the form (x-a)A with aD. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether Mult_1(A, . ) is dense in Mult_m(A, . ). 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.