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 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 amaximal ideal and let Mult₁(A, ‖. ‖) be the subset of the ϕ ∈ Mult(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 Mult₁(A, ‖. ‖) is dense in Mult_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 ℂ_p. On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult₁(A, ‖. ‖), which suggest that Mult₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).