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

Pestov, Vladimir

Comptes Rendus. Mathématique, Tome 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, $\pi$, d’un tel groupe (que nous disons d’être “moyennable en biais”) possède un état sur $B\left({\mathscr{H}}_{\pi }\right)$ 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 $\pi$ of such a group (which we call skew-amenable) has a conjugation-invariant state on $B\left({\mathscr{H}}_{\pi }\right)$.

[1] Albeverio, Sergio; Høegh-Krohn, Raphael; Marion, Jean A.; Testard, Daniel H.; Torrésani, Bruno S. Noncommutative distributions. Unitary representation of gauge groups and algebras, Pure and Applied Mathematics, 175, Marcel Dekker, 1993 | Zbl

[2] Bekka, Mohammed E. B. Amenable unitary representations of locally compact groups, Invent. Math., Volume 100 (1990) no. 2, pp. 383-401 | DOI | MR | Zbl

[3] Bourbaki, Nicolas Topologie Générale I, Hermann, 1971

[4] Carderi, Alessandro; Thom, Andreas An exotic group as limit of finite special linear groups, Ann. Inst. Fourier, Volume 68 (2018) no. 1, pp. 257-273 | DOI | Numdam | MR | Zbl

[5] Carey, Alan; Grundling, Hendrik On the problem of the amenability of the gauge group, Lett. Math. Phys., Volume 68 (2004) no. 2, pp. 113-120 | DOI | MR | Zbl

[6] De Vries, Jan Elements of topological dynamics, Mathematics and its Applications, 257, Kluwer Academic Publishers, 1993 | MR | Zbl

[7] Dollard, John D.; Friedman, Charles N. Product integration with applications to differential equations, Encyclopedia of Mathematics and its Applications, 10, Addison-Wesley Publishing Group, 1979 | MR | Zbl

[8] Engelking, Ryszard General Topology, Monografie Matematyczne, 60, PWN - Polish Scientific Publishers, 1977 | Zbl

[9] Fremlin, David H. Measure theory. Vol. 2. Broad foundations, Torres Fremlin, 2003 (corrected second printing of the 2001 original) | Zbl

[10] Garrido, M. Isabel; Jaramillo, Jesús A. Lipschitz-type functions on metric spaces, J. Math. Anal. Appl., Volume 340 (2008) no. 1, pp. 282-290 | DOI | MR | Zbl

[11] Giordano, Thierry; Pestov, Vladimir Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 279-315 | DOI | MR | Zbl

[12] Greenleaf, Frederick P. Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, 16, Van Nostrand Reinhold Co., 1969 | MR | Zbl

[13] de la Harpe, Pierre Moyennabilité de quelques groupes topologiques de dimension infinie, C. R. Math. Acad. Sci. Paris, Volume 277 (1973), pp. 1037-1040 | Zbl

[14] de la Harpe, Pierre Moyennabilité du groupe unitaire et propriété $P$ de Schwartz des algèbres de von Neumann, Algèbres d’opérateurs (Sém., Les Plains-sur-Bex, 1978) (Lecture Notes in Mathematics), Volume 725, Springer, 1979, pp. 220-227 | DOI | Zbl

[15] Ledoux, Michel The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, 2001 | MR | Zbl

[16] Malliavin, Marie-Paule; Malliavin, Paul Integration on loop group III. Asymptotic Peter–Weyl orthogonality, J. Funct. Anal., Volume 108 (1992), pp. 13-46 | DOI | MR | Zbl

[17] Omori, Hideki Infinite-dimensional Lie groups, Translations of Mathematical Monographs, 158, American Mathematical Society, 1997 | MR | Zbl

[18] Palais, Richard S. Foundations of global non-linear analysis, Mathematics Lecture Note Series, W.A. Benjamin, Inc., 1968 | Zbl

[19] Pestov, Vladimir Dynamics of Infinite-Dimensional Groups: The Ramsey–Dvoretzky–Milman Phenomenon, University Lecture Series, 40, American Mathematical Society, 2006 | Zbl

[20] Samuel, Pierre Ultrafilters and compactification of uniform spaces, Trans. Am. Math. Soc., Volume 64 (1948), pp. 100-132 | DOI | MR | Zbl

[21] Schneider, Friedrich Martin; Thom, Andreas On Følner sets in topological groups, Compos. Math., Volume 154 (2018) no. 7, pp. 1333-1361 | DOI | Zbl