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- Title
THE ULTRAMETRIC CORONA PROBLEM AND SPHERICALLY COMPLETE FIELDS.
- Authors
Escassut, Alain
- 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 Multm(A, ∥ · ∥) be the subset of the ϕ ϵ Mult(A,∥ · ∥) whose kernel is a maximal ideal and let Multa(A, ∥ · ∥) be the subset of the ϕ ϵ Mult(A, ∥ · ∥) whose kernel is a maximal ideal of the form (x - a)A with a ϕ ϵ 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 Multa(A, ∥ · ∥) dense in Multm(A,∥ · ∥)? Is it dense in Mult(A, ∥ · ∥)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique ϕ ϵ Multm(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 ϵ Multm(A, ∥ · ∥). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ∥ · ∥) in Multm(A, ∥ · ∥).
- Subjects
BANACH algebras; ERROR functions; POLYNOMIALS; KERNEL functions; ANALYTIC functions
- Publication
Proceedings of the Edinburgh Mathematical Society, 2010, Vol 53, Issue 2, p353
- ISSN
0013-0915
- Publication type
Article
- DOI
10.1017/S0013091508000837