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.
@article{urn:eudml:doc:44376,
title = {Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {13},
year = {2000},
pages = {85-109},
zbl = {0979.46053},
mrnumber = {MR1794904},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44376}
}
Boussaf, K.; Maïnetti, N.; Hemdaoui, M. Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D).. Revista Matemática de la Universidad Complutense de Madrid, Tome 13 (2000) pp. 85-109. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44376/