The Ultrametric Corona Problem and spherically complete fields

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 disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult_m(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Mult_a(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Mult_a(A, ‖ · ‖) dense in Mult_m(A, ‖ · ‖)? Is it dense in Mult_m(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(_m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(_m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Mult_a(A, ‖ · ‖) in Mult_m(A, ‖ · ‖).