# Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).

K. Boussaf; N. Maïnetti; M. Hemdaoui

Revista Matemática Complutense (2000)

- Volume: 13, Issue: 1, page 85-109
- ISSN: 1139-1138

## Access Full Article

top## Abstract

top## How to cite

topBoussaf, K., Maïnetti, N., and Hemdaoui, M.. "Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).." Revista Matemática Complutense 13.1 (2000): 85-109. <http://eudml.org/doc/44376>.

@article{Boussaf2000,

abstract = {Let K be an algebraically closed field, complete for an ultra- metric absolute value, let D be an infinite subset of K and let H(D) be the set of analytic elements on D. We denote by Mult(H(D), UD) the set of semi-norms Phi of the K-vector space H(D) which are continuous with respect to the topology of uniform convergence on D and which satisfy further Phi(f g)=Phi(f) Phi(g) whenever f,g elements of H(D) such that fg element of H(D). This set is provided with the topology of simple convergence. By the way of a metric topology thinner than the simple convergence, we establish the equivalence between the connectedness of Mult(H(D),UD), the arc-connectedness of Mult(H(D),UD) and the infraconnectedness of D. This generalizes a result of Berkovich given on affinoid algebras. Next, we study the filter of neighbourhoods of an element of Mult(H(D),UD), and we give a condition on the field K such that this filter admits a countable basis. We also prove the local arc-connectedness of Mult(H(D),UD) when D is infraconnected. Finally, we study the metrizability of the topology of simple convergence on Mult(H(D), UD) and we give some conditions to have an equivalence with the metric topology defined above. The fundamental tool in this survey consists of circular filters.},

author = {Boussaf, K., Maïnetti, N., Hemdaoui, M.},

journal = {Revista Matemática Complutense},

keywords = {Análisis no arquimediano; Seminormas; Espacios vectoriales topológicos; set of multiplicative seminorms; Krasner algebras; arc-connectedness; infra connectedness},

language = {eng},

number = {1},

pages = {85-109},

title = {Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).},

url = {http://eudml.org/doc/44376},

volume = {13},

year = {2000},

}

TY - JOUR

AU - Boussaf, K.

AU - Maïnetti, N.

AU - Hemdaoui, M.

TI - Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).

JO - Revista Matemática Complutense

PY - 2000

VL - 13

IS - 1

SP - 85

EP - 109

AB - Let K be an algebraically closed field, complete for an ultra- metric absolute value, let D be an infinite subset of K and let H(D) be the set of analytic elements on D. We denote by Mult(H(D), UD) the set of semi-norms Phi of the K-vector space H(D) which are continuous with respect to the topology of uniform convergence on D and which satisfy further Phi(f g)=Phi(f) Phi(g) whenever f,g elements of H(D) such that fg element of H(D). This set is provided with the topology of simple convergence. By the way of a metric topology thinner than the simple convergence, we establish the equivalence between the connectedness of Mult(H(D),UD), the arc-connectedness of Mult(H(D),UD) and the infraconnectedness of D. This generalizes a result of Berkovich given on affinoid algebras. Next, we study the filter of neighbourhoods of an element of Mult(H(D),UD), and we give a condition on the field K such that this filter admits a countable basis. We also prove the local arc-connectedness of Mult(H(D),UD) when D is infraconnected. Finally, we study the metrizability of the topology of simple convergence on Mult(H(D), UD) and we give some conditions to have an equivalence with the metric topology defined above. The fundamental tool in this survey consists of circular filters.

LA - eng

KW - Análisis no arquimediano; Seminormas; Espacios vectoriales topológicos; set of multiplicative seminorms; Krasner algebras; arc-connectedness; infra connectedness

UR - http://eudml.org/doc/44376

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.