[Sur les groupes de Lie connexes et la propriété d'approximation]
Une caractérisation complète des groupes de Lie connexes avec la propriété d'approximation a été obtenue récemment. La preuve utilisait la propriété (T⁎), nouvellement introduite. Nous présentons ici une preuve courte du même résultat sans utiliser la propriété (T⁎). En utilisant (T⁎), cependant, la caractérisation est étendue aux groupes localement compacts presque connexes. Nous concluons avec quelques remarques sur la difficulté d'aller au-delà du cas presque connexe.
Recently, a complete characterization of connected Lie groups with the Approximation Property was given. The proof used the newly introduced property (T⁎). We present here a short proof of the same result avoiding the use of property (T⁎). Using property (T⁎), however, the characterization is extended to almost connected locally compact groups. We end with some remarks about the difficulty of going beyond the almost connected case.
Accepté le :
Publié le :
@article{CRMATH_2016__354_7_697_0,
author = {Knudby, S{\o}ren},
title = {On connected {Lie} groups and the approximation property},
journal = {Comptes Rendus. Math\'ematique},
pages = {697--699},
publisher = {Elsevier},
volume = {354},
number = {7},
year = {2016},
doi = {10.1016/j.crma.2016.04.007},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.04.007/}
}
TY - JOUR AU - Knudby, Søren TI - On connected Lie groups and the approximation property JO - Comptes Rendus. Mathématique PY - 2016 SP - 697 EP - 699 VL - 354 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.04.007/ DO - 10.1016/j.crma.2016.04.007 LA - en ID - CRMATH_2016__354_7_697_0 ER -
Knudby, Søren. On connected Lie groups and the approximation property. Comptes Rendus. Mathématique, Tome 354 (2016) no. 7, pp. 697-699. doi : 10.1016/j.crma.2016.04.007. http://www.numdam.org/articles/10.1016/j.crma.2016.04.007/
[1] Graphical small cancellation groups with the Haagerup property, 2014 (preprint) | arXiv
[2] -Algebras and Finite-Dimensional Approximations, Grad. Stud. Math., vol. 88, American Mathematical Society, Providence, RI, USA, 2008
[3] Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math., Volume 96 (1989) no. 3, pp. 507-549
[4] Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math., Volume 107 (1985) no. 2, pp. 455-500
[5] L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. Fr., Volume 92 (1964), pp. 181-236
[6] Simple Lie groups without the approximation property, Duke Math. J., Volume 162 (2013) no. 5, pp. 925-964
[7] Simple Lie groups without the approximation property, Trans. Amer. Math. Soc., Volume 368 (2016) no. 6, pp. 3777-3809
[8] A complete characterization of connected Lie groups with the approximation property, Ann. Sci. Éc. Norm. Super., Volume 49 (2016) no. 4 (preprint, 2014) | arXiv
[9] Approximation properties for group -algebras and group von Neumann algebras, Trans. Amer. Math. Soc., Volume 344 (1994) no. 2, pp. 667-699
[10] Noncommutative -spaces without the completely bounded approximation property, Duke Math. J., Volume 160 (2011) no. 1, pp. 71-116
[11] Weak amenability of the universal covering group of , Math. Ann., Volume 288 (1990) no. 3, pp. 445-472
[12] Topological Transformation Groups, Interscience Publishers, New York–London, 1955
[13] The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2), Volume 52 (1950), pp. 606-636
[14] Small cancellation labellings of some infinite graphs and applications, 2014 (preprint) | arXiv
[15] Lie Groups, Lie Algebras, and Their Representations, Grad. Texts Math., vol. 102, Springer-Verlag, New York, 1984 (reprint of the 1974 edition)
Cité par Sources :
