Duke Mathematical Journal

Noncommutative Lp-spaces without the completely bounded approximation property

Vincent Lafforgue and Mikael De la Salle

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Abstract

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

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

Dates
First available in Project Euclid: 27 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1317149892

Digital Object Identifier
doi:10.1215/00127094-1443478

Mathematical Reviews number (MathSciNet)
MR2838352

Zentralblatt MATH identifier
1267.46072

Subjects
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]

Citation

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


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