An amenability-like property of finite energy path and loop groups

Vladimir Pestov

doi:10.5802/crmath.134

Group Theory

An amenability-like property of finite energy path and loop groups

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 11-12, pp. 1139-1155.

Abstract
Résumé

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 $B (ℋ_{π})$ .

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 $B (ℋ_{π})$ invariant sous conjugaison.

Received: 2020-08-12
Revised: 2020-10-20
Accepted: 2020-10-20
Published online: 2021-01-25

DOI: 10.5802/crmath.134

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Vladimir Pestov},
     title = {An amenability-like property of finite energy path and loop groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1139--1155},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2020},
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     language = {en},
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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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