Duke Mathematical Journal

Noncommutative Lp-spaces without the completely bounded approximation property

Vincent Lafforgue and Mikael De la Salle

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 any 1p different from 2, we give examples of noncommutative Lp-spaces without the completely bounded approximation property. Let F be a nonarchimedian local field. If p>4 or p<4/3 and r3 these examples are the noncommutative Lp-spaces of the von Neumann algebra of lattices in SLr(F) or in SLr(R). For other values of p the examples are the noncommutative Lp-spaces of the von Neumann algebra of lattices in SLr(F) for r large enough depending on p.

We also prove that if r3 lattices in SLr(F) or SLr(R) do not have the approximation property of Haagerup and Kraus. This provides examples of exact C-algebras without the operator space approximation property.

Article information

Duke Math. J., Volume 160, Number 1 (2011), 71-116.

First available in Project Euclid: 27 September 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Lafforgue, Vincent; De la Salle, Mikael. Noncommutative $L^{p}$ -spaces without the completely bounded approximation property. Duke Math. J. 160 (2011), no. 1, 71--116. doi:10.1215/00127094-1443478. https://projecteuclid.org/euclid.dmj/1317149892

Export citation


  • [1] N. Bourbaki, Éléments de mathématique, Fasc. XIII, Livre VI: Intégration, 2nd ed., Actualités Scientifiques et Industrielles 1175, Hermann, Paris, 1965.
  • [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] M. Bożejko and G. Fendler, Herz-Schur multipliers and uniformly bounded representations of discrete groups, Arch. Math. 57 (1991), 290–298.
  • [4] N. P. Brown and N. Ozawa, C-Algebras and Finite-Dimensional Approximations, Grad. Stud. Math. 88, Amer. Math. Soc., Providence, 2008.
  • [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] 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.
  • [7] J. Diestel and J. J. Uhl, jr, Vector Measures, Math. Surv. 15, Amer. Math. Soc., Providence, 1977.
  • [8] E. G. Effros and Z.-J. Ruan, Operator Spaces, London Math. Soc. Monogr. N.S. 23, Clarendon Press/Oxford Univ. Press, New York, 2000.
  • [9] E. Guentner, N. Higson, and S. Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243–268.
  • [10] U. Haagerup, Group C-algebras without the completely bounded approximation property, preprint, 1986.
  • [11] U. Haagerup and J. Kraus, Approximation properties for group C-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), 667–699.
  • [12] U. Haagerup, T. Steenstrup, and R. Szwarc, Schur multipliers and spherical functions on homogeneous trees, Internat. J. Math. 21 (2010), 1337–1382.
  • [13] M. Junge, Applications of Fubini’s theorem for noncommutative Lp spaces, preprint, 2004.
  • [14] M. Junge, N. J. Nielsen, Z.-J. Ruan, and Q. Xu, $\mathscr{C}\mathscr{O}\mathscr{L}_{p}$ spaces—the local structure of non-commutative Lp spaces, Adv. Math. 187 (2004), 257–319.
  • [15] M. Junge and Z.-J. Ruan, Approximation properties for noncommutative Lp-spaces associated with discrete groups, Duke Math. J. 117 (2003), 313–341.
  • [16] E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group C-algebras, Invent. Math. 112 (1993), 449–489.
  • [17] E. Kissin and V. S. Shulman, Operator multipliers, Pacific J. Math. 227 (2006), 109–141.
  • [18] V. Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008), 559–602.
  • [19] 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.
  • [20] 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 3 August 2011).
  • [21] V. Losert, On multipliers and completely bounded multipliers—The case SL(2, R), preprint, 2010.
  • [22] S. Neuwirth and E. Ricard, Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group, to appear in Canad. J. Math., preprint, arXiv:1001.5332v1 [math.FA]
  • [23] N. Ozawa, About the QWEP conjecture, Internat. J. Math. 15 (2004), 501–530.
  • [24] G. Pisier, Regular operators between non-commutative Lp-spaces, Bull. Sci. Math. 119 (1995), 95–118.
  • [25] G. Pisier, Non-Commutative Vector Valued Lp-Spaces and Completely p-Summing Maps, Astérisque 247, Soc. Math. France, Montrouge, 1998.
  • [26] G. Pisier, Similarity Problems and Completely Bounded Maps, 2nd expanded ed., Lecture Notes in Math. 1618, Springer, Berlin, 2001.
  • [27] G. Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser. 294, Cambridge Univ. Press, Cambridge, 2003.
  • [28] A. Szankowski, On the uniform approximation property in Banach spaces, Israel J. Math. 49 (1984), 343–359.