A complete characterization of connected Lie groups with the Approximation Property

Haagerup, Uffe; Knudby, Søren; de Laat, Tim

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 927-946.

Nous donnons une caractérisation complète des groupes de Lie connexes ayant la propriété d'approximation (AP) pour des groupes. À cette fin, nous introduisons un renforcement de la propriété (T), que nous appelons propriété (T${}^{*}$) et qui est une obstruction naturelle à AP. Dans le but de définir la propriété (T${}^{*}$), nous montrons d'abord que pour tout groupe localement compact $G$, l'espace $M_{0}A\left(G\right)$ des multiplicateurs complètement bornés de $G$ admet une unique moyenne invariante à gauche $m$. Un groupe localement compact $G$ a la propriété (T${}^{*}$) si $m$ est une forme continue pour la topologie ${}^{*}$-faible. Après avoir démontré que les groupes $\mathrm{SL}(3,ℝ)$, $\mathrm{Sp}(2,ℝ)$ et $\widetilde{\mathrm{Sp}}(2,ℝ)$ ont la propriété (T${}^{*}$), nous étudions la question de savoir lesquels parmi les groupes de Lie connexes ont l'AP. Il se pose alors le problème technique que la partie semi-simple de la décomposition de Levi globale d'un groupe de Lie connexe n'est pas toujours fermée. Grâce à une importante propriété de stabilité de la propriété (T${}^{*}$), ce problème disparaît. Il s'en suit qu'un groupe de Lie connexe a l'AP si et seulement si tous les facteurs simples de la partie semi-simple de sa décomposition de Levi ont un rang réel 0 ou 1. Enfin, nous démontrons que tous les groupes de Lie simples connexes de rang $\ge 2$ et de centre fini ont la propriété (T${}^{*}$).

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T${}^{*}$), which is a natural obstruction to the AP. In order to define property (T${}^{*}$), we first prove that for every locally compact group $G$, there exists a unique left invariant mean $m$ on the space $M_{0}A\left(G\right)$ of completely bounded Fourier multipliers of $G$. A locally compact group $G$ is said to have property (T${}^{*}$) if this mean $m$ is a weak${}^{*}$ continuous functional. After proving that the groups $\mathrm{SL}(3,ℝ)$, $\mathrm{Sp}(2,ℝ)$, and $\widetilde{\mathrm{Sp}}(2,ℝ)$ have property (T${}^{*}$), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T${}^{*}$), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T${}^{*}$) for all connected simple higher rank Lie groups with finite center.

Classification : 22D25, 46B28.
Keywords: Approximation properties, Lie groups, property (T), invariant means.
Mot clés : Propriétés d'approximation, groupes de Lie, propriété (T), moyennes invariantes.

Akemann, C. A.; Walter, M. E. Unbounded negative definite functions, Canad. J. Math., Volume 33 (1981), pp. 862-871 (ISSN: 0008-414X) | DOI | MR | Zbl

Bekka, B.; de la Harpe, P.; Valette, A., Cambridge Univ. Press, 2008 | MR | Zbl

Bożejko, M.; Fendler, G. Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A, Volume 3 (1984), pp. 297-302 | MR | Zbl

Bożejko, M. Positive and negative definite kernels on discrete groups (1987) (preprint lecture notes at Heidelberg University)

Borel, A.; Tits, J. Groupes réductifs, Publ. Math. IHÉS, Volume 27 (1965), pp. 55-150 (ISSN: 0073-8301) | DOI | Numdam | MR

Burckel, R. B., Gordon and Breach Science Publishers, 1970, 118 pages | MR | Zbl

Cherix, P.-A.; Cowling, M.; Jolissaint, P.; Julg, P.; Valette, A., Modern Birkhäuser Classics, Birkhäuser, 2001, 126 pages (ISBN: 978-3-0348-0905-4; 978-3-0348-0906-1) | DOI | MR | Zbl

Cowling, M.; Dorofaeff, B.; Seeger, A.; Wright, J. A family of singular oscillatory integral operators and failure of weak amenability, Duke Math. J., Volume 127 (2005), pp. 429-486 (ISSN: 0012-7094) | DOI | MR | Zbl

Cowling, M.; Haagerup, U. Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. math., Volume 96 (1989), pp. 507-549 (ISSN: 0020-9910) | DOI | MR | Zbl

De Cannière, J.; Haagerup, U. Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math., Volume 107 (1985), pp. 455-500 (ISSN: 0002-9327) | DOI | MR | Zbl

de Leeuw, K.; Glicksberg, I. Almost periodic compactifications, Bull. Amer. Math. Soc., Volume 65 (1959), pp. 134-139 (ISSN: 0002-9904) | DOI | MR | Zbl

de Leeuw, K.; Glicksberg, I. Applications of almost periodic compactifications, Acta Math., Volume 105 (1961), pp. 63-97 (ISSN: 0001-5962) | DOI | MR | Zbl

Dorofaeff, B. Weak amenability and semidirect products in simple Lie groups, Math. Ann., Volume 306 (1996), pp. 737-742 (ISSN: 0025-5831) | DOI | MR | Zbl

Eberlein, W. F. Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., Volume 67 (1949), pp. 217-240 (ISSN: 0002-9947) | DOI | MR | Zbl

Effros, E. G.; Ruan, Z.-J., London Mathematical Society Monographs. New Series, 23, The Clarendon Press, Oxford Univ. Press, 2000, 363 pages (ISBN: 0-19-853482-5) | MR | Zbl

Eymard, P. L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France, Volume 92 (1964), pp. 181-236 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Eymard, P., Applications of hypergroups and related measure algebras (Seattle, WA, 1993) (Contemp. Math.), Volume 183, Amer. Math. Soc., Providence, RI, 1995, pp. 111-128 | DOI | MR | Zbl

Godement, R. Les fonctions de type positif et la théorie des groupes, Trans. Amer. Math. Soc., Volume 63 (1948), pp. 1-84 (ISSN: 0002-9947) | MR | Zbl

Greenleaf, F. P., Van Nostrand Mathematical Studies, 16, Van Nostrand Reinhold Co., 1969, 113 pages | MR | Zbl

Grothendieck, A. Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., Volume 16 (1955) (ISSN: 0065-9266) | MR | Zbl

Haagerup, U.; de Laat, T. Simple Lie groups without the Approximation Property, Duke Math. J., Volume 162 (2013), pp. 925-964 (ISSN: 0012-7094) | DOI | MR | Zbl

Haagerup, U.; de Laat, T. Simple Lie groups without the Approximation Property II, Trans. Amer. Math. Soc., Volume 368 (2016), pp. 3777-3809 (ISSN: 0002-9947) | DOI | MR | Zbl

Helgason, S., Pure and Applied Mathematics, 80, Academic Press, Inc., 1978, 628 pages (ISBN: 0-12-338460-5) | MR | Zbl

Herz, C. Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble), Volume 24 (1974), pp. 145-157 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Haagerup, U.; Kraus, J. Approximation properties for group $C^{*}$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc., Volume 344 (1994), pp. 667-699 (ISSN: 0002-9947) | DOI | MR | Zbl

Každan, D. A. On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., Volume 1 (1967), pp. 71-74 (ISSN: 0374-1990) | MR | Zbl

Knapp, A. W., Progress in Math., 140, Birkhäuser, 2002, 812 pages (ISBN: 0-8176-4259-5) | MR | Zbl

Lafforgue, V.; de la Salle, M. Noncommutative $L^p$-spaces without the completely bounded approximation property, Duke Math. J., Volume 160 (2011), pp. 71-116 (ISSN: 0012-7094) | DOI | MR | Zbl

Mostow, G. D. The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math., Volume 52 (1950), pp. 606-636 (ISSN: 0003-486X) | DOI | MR | Zbl

Pier, J.-P., Pure and Applied Mathematics, John Wiley & Sons, Inc., 1984, 418 pages (ISBN: 0-471-89390-0) | MR | Zbl

Pisier, G., London Mathematical Society Lecture Note Series, 294, Cambridge Univ. Press, 2003, 478 pages (ISBN: 0-521-81165-1) | DOI | MR | Zbl

Rudin, W., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., 1991, 424 pages (ISBN: 0-07-054236-8) | MR | Zbl

Runde, V., Lecture Notes in Math., 1774, Springer, 2002, 296 pages (ISBN: 3-540-42852-6) | DOI | MR | Zbl

Valette, A. Minimal projections, integrable representations and property $\left(\mathrm{T}\right)$ , Arch. Math. (Basel), Volume 43 (1984), pp. 397-406 (ISSN: 0003-889X) | DOI | MR | Zbl

Varadarajan, V. S., Graduate Texts in Math., 102, Springer, 1984, 430 pages (ISBN: 0-387-90969-9) | DOI | MR | Zbl

Veech, W. A. Weakly almost periodic functions on semisimple Lie groups, Monatsh. Math., Volume 88 (1979), pp. 55-68 (ISSN: 0026-9255) | DOI | MR | Zbl

Xu, G. Herz-Schur multipliers and weakly almost periodic functions on locally compact groups, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 2525-2536 (ISSN: 0002-9947) | DOI | MR | Zbl