Duke Mathematical Journal

Simple Lie groups without the approximation property

Uffe Haagerup and Tim de Laat

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a locally compact group G, let A(G) denote its Fourier algebra, and let M0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-∗ topology on the space M0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely, SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.

Article information

Duke Math. J., Volume 162, Number 5 (2013), 925-964.

First available in Project Euclid: 29 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
Secondary: 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] 46L07: Operator spaces and completely bounded maps [See also 47L25]


Haagerup, Uffe; de Laat, Tim. Simple Lie groups without the approximation property. Duke Math. J. 162 (2013), no. 5, 925--964. doi:10.1215/00127094-2087672. https://projecteuclid.org/euclid.dmj/1364562915

Export citation


  • [1] A. Borel and J. Tits, Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–150.
  • [2] M. Bożejko and G. Fendler, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. A (6) 3 (1984), 297–302.
  • [3] N. P. Brown and N. Ozawa, $\operatorname{C}^{*}$-Algebras and Finite-Dimensional Approximations, Grad. Stud. Math. 88, Amer. Math. Soc., Providence, 2000.
  • [4] D. Bump, Lie Groups, Grad. Texts Math. 225, Springer, New York, 2004.
  • [5] J. de Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math 107 (1985), 455–500.
  • [6] J. Conway, A Course in Functional Analysis, Grad. Texts Math. 96, Springer, New York, 1990.
  • [7] M. Cowling, B. Dorofaeff, A. Seeger, and J. Wright, A family of singular oscillatory integral operators and failure of weak amenability, Duke Math. J. 127 (2005), 429–485.
  • [8] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507–549.
  • [9] G. van Dijk, Introduction to Harmonic Analysis and Generalized Gelfand Pairs, Stud. Math. 36, de Gruyter, Berlin, 2009.
  • [10] B. Dorofaeff, Weak amenability and semidirect products in simple Lie groups, Math. Ann. 306 (1996), 737–742.
  • [11] E. Effros and Z.-J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), 163–187.
  • [12] P. Eymard, L’algébre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236.
  • [13] P. Eymard, “A survey of Fourier algebras” in Applications of Hypergroups and Related Measure Algebras (Seattle, WA, 1993), Contemp. Math. 183, Amer. Math. Soc., Providence, 1995, 111–128.
  • [14] J. Faraut, “Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques” in Analyse Harmonique (Université de Nancy I, 1980), Les Cours du CIMPA, Nice, 1983, 315–446.
  • [15] J. Faraut, Analysis on Lie Groups, Cambridge Stud. Adv. Math. 110, Cambridge Univ. Press, Cambridge, 2008.
  • [16] E. Guentner, N. Higson, and S. Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243–268.
  • [17] U. Haagerup, Group $\operatorname{C}^{*}$-algebras without the completely bounded approximation property, unpublished manuscript, 1986.
  • [18] U. Haagerup and J. Kraus, Approximation properties for group $\operatorname{C}^{*}$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), 667–699.
  • [19] U. Haagerup and H. Schlichtkrull, Inequalities for Jacobi polynomials, to appear in Ramanujan J., preprint, arXiv:1201.0495v2 [math.RT].
  • [20] M. L. Hansen, Weak amenability of the universal covering group of SU(1,n), Math. Ann. 288 (1990), 445–472.
  • [21] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978.
  • [22] C. S. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), 145–157.
  • [23] A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, Boston, 1996.
  • [24] T. Koornwinder, The Addition Formula for Jacobi Polinomials, II in The Laplace Type Integral Representation and the Product Formula, Report TW 133/72, Mathematical Centre, Amsterdam 1972.
  • [25] V. Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008), 559–602.
  • [26] V. Lafforgue, “Propriété (T) renforcée et conjecture de Baum-Connes” in Quanta of Maths, Clay Math. Proc. 11, Amer. Math. Soc., Providence, 2010, 323–345.
  • [27] V. Lafforgue, Un analogue non archimédien d’un résultat de Haagerup et lien avec la propriété (T) renforcée, preprint, 2010, http://people.math.jussieu.fr/~vlafforg/haagerup-rem.pdf (accessed 15 January 2011).
  • [28] V. Lafforgue and M. de la Salle, Noncommutative $L^{p}$-spaces without the completely bounded approximation property, Duke. Math. J. 160 (2011), 71–116.
  • [29] H. Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A–B 266 (1968), 1180–1182.
  • [30] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, Berlin, 1991.
  • [31] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, 1939.
  • [32] S. P. Wang, The dual space of semi-simple Lie groups, Amer. J. Math. 91 (1969), 921–937.
  • [33] H. Weyl, The Theory of Groups and Quantum Mechanics, Methuen and Co., London, 1931.