Généralisation d'un théorème de Haagerup

Volume 168 / 2005

Ferdaous Kellil, Guy Rousseau Studia Mathematica 168 (2005), 217-227 MSC: 43A85, 20E08, 47A30. DOI: 10.4064/sm168-3-3

Abstract

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 $S$.

Authors

  • Ferdaous KellilDépartement de Mathématiques
    Faculté des Sciences de Monastir
    5000 Monastir, Tunisie
    e-mail
  • Guy RousseauInstitut Elie Cartan
    Unité mixte de Recherche 7502
    Université Henri Poincaré Nancy 1
    B.P. 239
    54506 Vandœuvre-lès-Nancy, France
    e-mail

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