Survey on the Kakutani problem in p-adic analysis II - Escassut, Alain (Sarajevo Journal of Mathematics, 2022)
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 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, . ) is dense in Mult_m(A, . ), beginning by the case when is spherically complete and generalizing next.