The Corona problem on a complete ultrametric algebraically closed field

Abstract

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the "open" unit disk D of IK provided with the Gauss norm. Let M ult(A, .) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let M ult m (A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a maximal ideal and let M ult 1 (A, .) be the subset of the φ ∈ M ult(A, .) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether M ult 1 (A, .) is dense in M ult m (A, .). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to l C p. On the other hand, we also show that the continuous multiplicative semi-norms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of M ult 1 (A, .), which suggest that M ult 1 (A, .) might be dense in M ult(A, .).

Nonmaximal ideals and the Berkovich space of the algebra of bounded analytic functions

“…Nevertheless, in our case, it is unknown if D is dense in M = M (H ∞ ), which is a nonarchimedean version of the Corona problem (a related problem was solved in [14]). In fact, what is now known is that D is dense in the subset of all seminorms whose kernel is a maximal ideal (see [7]). …”