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.
Type de document :
Article dans une revue
Duke Mathematical Journal, Duke University Press, 2011, 160 (1), pp.71-116. 〈10.1215/00127094-1443478〉
Domaine :

https://hal.archives-ouvertes.fr/hal-01144654
Contributeur : Vincent Lafforgue <>
Soumis le : mercredi 22 avril 2015 - 12:35:38
Dernière modification le : vendredi 6 juillet 2018 - 15:18:04

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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