# Noncommutative $L^{p}$ -spaces without the completely bounded approximation property

Abstract : For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for r large enough depending on p. We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have the Approximation Property of Haagerup and Kraus. This provides examples of exact C^*-algebras without the operator space approximation property.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01144654
Contributor : Vincent Lafforgue <>
Submitted on : Wednesday, April 22, 2015 - 12:35:38 PM
Last modification on : Friday, July 6, 2018 - 3:18:04 PM

### Citation

Vincent Lafforgue, Mikael de la Salle. Noncommutative $L^{p}$ -spaces without the completely bounded approximation property. Duke Mathematical Journal, Duke University Press, 2011, 160 (1), pp.71-116. ⟨10.1215/00127094-1443478⟩. ⟨hal-01144654⟩

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