Comptes Rendus
Théorie des groupes
An amenability-like property of finite energy path and loop groups
[Une propriété semblable à la moyennabilité des groupes de chemins et de lacets à énergie finie]
Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1139-1155.

Nous montrons que les groupes de lacets et de chemins à énergie finie (c.à.d. de classe H 1 de Sobolev) à valeurs dans un groupe de Lie compact et connexe, ainsi que leurs extensions centrales, satisfont une version de la moyennabilité : ils admettent une moyenne invariante à gauche sur l’espace de fonctions bornées uniformément continues par rapport a une métrique invariante à gauche. Chaque représentation unitaire continue, π, d’un tel groupe (que nous disons d’être “moyennable en biais”) possède un état sur B( π ) invariant sous conjugaison.

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B( π ).

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DOI : 10.5802/crmath.134

Vladimir Pestov 1, 2

1 Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Universitário Trindade, CEP 88.040-900 Florianópolis-SC, Brasil
2 Département de mathématiques et de statistique, Université d’Ottawa, Complexe STEM, 150 Louis-Pasteur Pvt, Ottawa, Ontario K1N 6N5 Canada
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Vladimir Pestov. An amenability-like property of finite energy path and loop groups. Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1139-1155. doi : 10.5802/crmath.134. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.134/

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