Geodesic
Amenability of Closed Subgroups and Orlicz Spaces
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 583-590.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that a closed subgroup $H$ of a second countable locally compact group $G$ is amenable if and only if its left regular representation on an Orlicz space $L^\Phi(G)$ for some $\Delta_2$-regular $N$-function $\Phi$ almost has invariant vectors. We also show that a noncompact second countable locally compact group $G$ is amenable if and ony if the first cohomology space $H^1(G,L^\Phi(G))$ is non-Hausdorff for some $\Delta_2$-regular $N$-function $\Phi$.
Keywords: locally compact group, second countable group, closed subgroup, $N$-function, Orlicz space, 1-cohomology.
Mots-clés : amenable group 
@article{SEMR_2013_10_a32,
     author = {Ya A. Kopylov},
     title = {Amenability of {Closed} {Subgroups} and {Orlicz} {Spaces}},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {583--590},
     publisher = {mathdoc},
     volume = {10},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2013_10_a32/}
}
TY  - JOUR
AU  - Ya A. Kopylov
TI  - Amenability of Closed Subgroups and Orlicz Spaces
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2013
SP  - 583
EP  - 590
VL  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2013_10_a32/
LA  - en
ID  - SEMR_2013_10_a32
ER  - 
%0 Journal Article
%A Ya A. Kopylov
%T Amenability of Closed Subgroups and Orlicz Spaces
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2013
%P 583-590
%V 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2013_10_a32/
%G en
%F SEMR_2013_10_a32
Ya A. Kopylov. Amenability of Closed Subgroups and Orlicz Spaces. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 583-590. http://geodesic.mathdoc.fr/item/SEMR_2013_10_a32/

[1] I. Akbarbaglu, S. Maghsoudi, “On certain porous sets in the Orlicz space of a locally compact group”, Colloq. Math., 129:1 (2012), 99–111 | DOI | MR | Zbl

[2] I. Akbarbaglu, S. Maghsoudi, “Banach–Orlicz algebras on a locally compact group”, Mediterr. J. Math., 2013 | DOI

[3] N. Bourbaki, Intégration, Chapitres VII, VIII, Act. Sci. Ind., 1306, Hermann, Paris, 1963 | MR | Zbl

[4] M. Bourdon, F. Martin, A. Valette, “Vanishing and non-vanishing for the first $L^p$-cohomology of groups”, Comm. Math. Helv., 80:2 (2005), 377–389 | DOI | MR | Zbl

[5] I. M. Bund, “Birbaum–Orlicz spaces of functions on groups”, Pac. J. Math., 58 (1975), 351–359 | DOI | MR | Zbl

[6] J. Dixmier, “Dual et quasi-dual d'une algebre de Banach involutive”, Trans. Amer. Math. Soc., 104:2 (1962), 278–283 | MR | Zbl

[7] P. Eymard, Moyennes Invariantes et Représentations Unitaires, Lect. Notes in Math., 300, Springer-Verlag, 1972, MR0447969 pp. | DOI | MR | Zbl

[8] A. Guichardet, “Sur la cohomologie des groupes topologiques, II”, Bull. Sci. Math. II, 96 (1972), 305–332 | MR | Zbl

[9] E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, v. I, 2nd ed., Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR | Zbl

[10] A. Kamińska, J. Musielak, “On convolution operator in Orlicz spaces”, Rev. Mat. Univ. Complutense Madr., 2, Suppl. (1989), 157–178 | MR | Zbl

[11] Ya. Kopylov, “An $L_p$-criterion of amenability for a locally compact group”, Sib. Èlektron. Mat. Izv., 2 (2005), 186–189 | MR | Zbl

[12] M. A. Krasnosel'skiĭ, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961 | MR

[13] F. Martin, A. Valette, “On the first $L^p$-cohomology of discrete groups”, Groups Geom. Dyn., 1:1 (2007), 81–100 | DOI | MR | Zbl

[14] M. M. Rao, “Convolutions of vector fields. II: Random walk models”, Nonlinear Anal., Theory Methods Appl., 47:6 (2001), 3599-3615 | DOI | MR | Zbl

[15] M. M. Rao, “Convolutions of vector fields. III: Amenability and spectral properties”, Real and stochastic analysis. New perspectives, Trends in Mathematics, ed. Rao M. M., Birkhäuser, Boston, MA, 2004, 375–401 | MR | Zbl

[16] M. M. Rao, Measure Theory and Integration, Pure and Applied Mathematics, 265, Marcel Dekker, New York, NY, 2004 | MR | Zbl

[17] M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York etc., 1991 | MR | Zbl

[18] M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, Pure and Applied Mathematics, 250, Marcel Dekker, New York, NY, 2002 | MR | Zbl

[19] H. Reiter, J. D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups, 2nd ed., Clarendon Press, Oxford, 2000 | MR | Zbl

[20] J. D. Stegeman, “On a property concerning locally compact groups”, Nederl. Akad. Wet., Proc., Ser. A, 68 (1965), 702–703 | MR | Zbl