Analytic mappings in the tree Mult(K[x]).

Kamal Boussaf, Alain Escassut, and Nicolas Maïnetti

Abstract     Let K be an algebraically closed complete ultrametric field, let $D\subset K$ be closed and bounded, and let H(D) be the Banach K-algebra of analytic elements on D. Let Mult(K[x]) (resp. $Mult(H(D), \Vert \ . \ \Vert_D)$) be the set of multiplicative semi-norms on K[x]) (resp. of continuous multiplicative semi-norms on H(D)) which are known to be characterized by circular filters. Mult(K[x]) is provided with the topology of simple convergence, and with a metric topology based upon a tree structure for which it is complete. Given a bounded closed infraconnected set $D\subset K$, the boundary of $Mult(H(D),\Vert \ . \ \Vert)$ inside Mult(K[x]) with respect to the topology of simple convergence, is equal to the Shilov boundary for $(H(D),\Vert \ . \ \Vert)$. If D is affinoid (particularly), this is also the boundary of $Mult(H(D),\Vert \ . \ \Vert)$ inside Mult(K[x]) with respect to the metric topology. We show that every element $f\in H(D)$ has continuation to a mapping f^* from $Mult(H(D), \Vert \ . \ \Vert_D)$ to Mult(K[x]) which is continuous for both topologies and uniformly continuous for the metric topology. The family of functions $Z_{\cal F}$ from H(D) to Mult(K[x]) defined as $Z_{\cal F}(f)=f^*({\cal F})$ (where $\cal F$ is a circular filter secant with D) is uniformly equicontinuous with respect to the metric topology. If the field K is separable, f^* is uniformly continuous for both topologies. Results also apply to meromorphic functions in K. A meromorphic function in K defines an increasing function f^* (with respect to the order of Mult(K[x])) if and only if it is an entire function. In a Krasner-Tate algebra $H(D)=K\{t\}[x]$, where $K\{t\}$ is a topologically pure extension of dimension 1 and x is the identical function on D integral over $K\{t\}$, we can precisely show that the Gauss norm on $K\{t\}$ admits a number of extensions to $K\{t\}[x]$ which is equal to the cardinal of the Shilov boundary of $Mult(H(D), \Vert \ . \ \Vert_D)$.


AMS Classification: 46S10, 12J25, 12J27