Généralisation d'un théorème de Haagerup
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 $S$.