Généralisation d'un théorème de Haagerup

Ferdaous Kellil ¹   ; Guy Rousseau ²

¹ Département de Mathématiques Faculté des Sciences de Monastir 5000 Monastir, Tunisie
² Institut Elie Cartan Unité mixte de Recherche 7502 Université Henri Poincaré Nancy 1 B.P. 239 54506 Vandœuvre-lès-Nancy, France

Studia Mathematica, Tome 168 (2005) no. 3, pp. 217-227

Résumé

Let $G$ be a group of automorphisms of a tree $X$ (with set of vertices $S$) and $H$ a kernel on $S\times S$ invariant under the action of $G$. We want to give an estimate of the $l^r$-operator norm $(1\leq r\leq 2)$ of the operator associated to $H$ in terms of a norm for $H$. This was obtained by U. Haagerup when $G$ is the free group acting simply transitively on a homogeneous tree. Our result is valid when $X$ is a locally finite tree and one of the orbits of $G$ is the set of vertices at even distance from a given vertex; a technical hypothesis, always true when $G$ is discrete, is also assumed. As an application we prove the invertibility of an $l^r$-operator on $S$.

Détail

DOI : 10.4064/sm168-3-3

Mots-clés : group automorphisms tree set vertices kernel times invariant under action want estimate r operator norm leq leq operator associated terms norm obtained haagerup group acting simply transitively homogeneous tree result valid locally finite tree orbits set vertices even distance given vertex technical hypothesis always discrete assumed application prove invertibility r operator