Fixed points for bounded orbits  in Hilbert spaces

Gheysens, Maxime; Monod, Nicolas

doi:10.24033/asens.2317

Fixed points for bounded orbits in Hilbert spaces
[Points fixes en présence d'orbites bornées dans les espaces hilbertiens]

Gheysens, Maxime ; Monod, Nicolas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 131-156.

Résumé
Abstract

Nous considérons la propriété suivante pour un groupe topologique $G$ : toute action affine continue de $G$ sur un espace hilbertien ayant une orbite bornée a un point fixe. Nous montrons qu'elle caractérise la moyennabilité des groupes localement compacts dénombrables à l'infini (en particulier des groupes discrets dénombrables).

Pour ce faire, nous introduisons une variante « modérée » de l'induction des représentations et nous généralisons le théorème de Gaboriau-Lyons pour montrer que tout groupe localement compact non moyennable admet, dans un sens probabiliste, des sous-groupes libres discrets. Ceci fournit une « solution au sens de la mesure » au problème de von Neumann pour les groupes localement compacts.

Nous illustrons ce dernier résultat en fournissant une réponse partielle au problème de Dixmier pour les groupes localement compacts.

Consider the following property of a topological group $G$ : every continuous affine $G$ -action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact $σ$ -compact groups (e.g., countable groups).

Along the way, we introduce a “moderate” variant of the classical induction of representations and we generalize the Gaboriau-Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the “measure-theoretic solution” to the von Neumann problem for locally compact groups.

We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.

Publié le : 2017-11-12

MR Zbl

DOI : 10.24033/asens.2317

Classification : 47H10, 22D12, 22A05, 43A07, 20E05, 37A20
Keywords: Amenable group, fixed point theorem, von Neumann problem, Dixmier problem
Mot clés : Groupe moyennable, théorème du point fixe, problème de von Neumann, problème de Dixmier

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     author = {Gheysens, Maxime and Monod, Nicolas},
     title = {Fixed points for bounded orbits  in {Hilbert} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {131--156},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {1},
     year = {2017},
     doi = {10.24033/asens.2317},
     mrnumber = {3621428},
     zbl = {1373.43001},
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     url = {http://www.numdam.org/articles/10.24033/asens.2317/}
}

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Gheysens, Maxime; Monod, Nicolas. Fixed points for bounded orbits  in Hilbert spaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 131-156. doi : 10.24033/asens.2317. http://www.numdam.org/articles/10.24033/asens.2317/

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