A complete characterization of connected Lie groups with the Approximation Property

Haagerup, Uffe; Knudby, Søren; de Laat, Tim

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 927-946.

Nous donnons une caractérisation complète des groupes de Lie connexes ayant la propriété d'approximation (AP) pour des groupes. À cette fin, nous introduisons un renforcement de la propriété (T), que nous appelons propriété (T*) et qui est une obstruction naturelle à AP. Dans le but de définir la propriété (T*), nous montrons d'abord que pour tout groupe localement compact G, l'espace M0A(G) des multiplicateurs complètement bornés de G admet une unique moyenne invariante à gauche m. Un groupe localement compact G a la propriété (T*) si m est une forme continue pour la topologie *-faible. Après avoir démontré que les groupes SL (3,ℝ), Sp (2,ℝ) et Sp ˜(2,ℝ) ont la propriété (T*), nous étudions la question de savoir lesquels parmi les groupes de Lie connexes ont l'AP. Il se pose alors le problème technique que la partie semi-simple de la décomposition de Levi globale d'un groupe de Lie connexe n'est pas toujours fermée. Grâce à une importante propriété de stabilité de la propriété (T*), ce problème disparaît. Il s'en suit qu'un groupe de Lie connexe a l'AP si et seulement si tous les facteurs simples de la partie semi-simple de sa décomposition de Levi ont un rang réel 0 ou 1. Enfin, nous démontrons que tous les groupes de Lie simples connexes de rang ≥2 et de centre fini ont la propriété (T*).

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean m on the space M0A(G) of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean m is a weak* continuous functional. After proving that the groups SL (3,ℝ), Sp (2,ℝ), and Sp ˜(2,ℝ) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.

Classification : 22D25, 46B28.
Keywords: Approximation properties, Lie groups, property (T), invariant means.
Mot clés : Propriétés d'approximation, groupes de Lie, propriété (T), moyennes invariantes.

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