The ultrametric corona problem

Abstract : Let $K$ be a complete ultrametric algebraically closed field and let $A$ be the $K$-Banach algebra of bounded analytic functions in the disk $D:\ |x|<1$. Let $Mult(A,\Vert \ . \ \Vert)$ be the set of continuous multiplicative semi-norms of $A$, let $Mult_m(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult(A,\Vert \ . \ \Vert)$ whose kernel is a maximal ideal and let $Mult_a(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult_m(A,\Vert \ . \ \Vert)$ whose kernel is of the form $(x-a)A, \ a\in D$ ( if $\phi \in Mult_m(A,\Vert \ . \ \Vert)\setminus Mult_a(A,\Vert \ . \ \Vert)$, the kernel of $\phi$ is then of infinite codimension). The main problem we examine is whether $Mult_a(A,\Vert \ . \ \Vert)$ is dense inside $Mult_m(A,\Vert \ . \ \Vert)$ with respect to the topology of simple convergence. This a first step to the conjecture of density of $Mult_a(A,\Vert \ . \ \Vert)$ in the whole set $Mult(A,\Vert \ . \ \Vert)$: this is the corresponding problem to the well-known complex corona problem. We notice that if $\phi \in Mult_m(A,\Vert \ . \ \Vert)$ is defined by an ultrafilter on $D$, $\phi$ lies in the closure of $Mult_a(A,\Vert \ . \ \Vert)$. Particularly, we shaw that this is case when a maximal ideal is the kernel of a unique $\phi \in Mult_m(A,\Vert \ . \ \Vert)$. Thus, if every maximal ideal is the kernel of a unique $\phi \in Mult_m(A,\Vert \ . \ \Vert)$, $Mult_a(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$. And particularly, this is the case when $K$ is strongly valued. In the general context, we find a subset of $Mult_m(A,\Vert \ . \ \Vert)\setminus Mult_a(A,\Vert \ . \ \Vert)$ which is included in the closure of $Mult_a(A,\Vert \ . \ \Vert)$. More generally, we show that if $\psi \in Mult(A,\Vert \ . \ \Vert)$ does not define the Gauss norm on polynomials $(\Vert \ . \ \Vert)$, then it is characterized by a circular filter, like on rational functions and analytic elements. As a consequence, if $\psi$ does not lie in the closure of $Mult_a(A,\Vert \ . \ \Vert)$, then its restriction to polynomials is the Gauss norm.
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Contemporary mathematics, American Mathematical Society, 2010, 508, pp.35-45
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Alain Escassut, Nicolas Mainetti. The ultrametric corona problem. Contemporary mathematics, American Mathematical Society, 2010, 508, pp.35-45. 〈hal-00681913〉

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