## Duke Mathematical Journal

### 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 noncommutative $L^{p}$-spaces without the completely bounded approximation property. Let $F$ be a nonarchimedian local field. If $p\textgreater 4$ or $p\textless 4/3$ and $r\geq3$ these examples are the noncommutative $L^{p}$-spaces of the von Neumann algebra of lattices in $\mathrm{S}\mathrm{L}_{r}(F)$ or in $\mathrm{S}\mathrm{L}_{r}({\mathbb{R}})$. For other values of $p$ the examples are the noncommutative $L^{p}$-spaces of the von Neumann algebra of lattices in $\mathrm{S}\mathrm{L}_{r}(F)$ for $r$ large enough depending on $p$.

We also prove that if $r\geq3$ lattices in $\mathrm{S}\mathrm{L}_{r}(F)$ or $\mathrm{S}\mathrm{L}_{r}({\mathbb{R}})$ 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

https://projecteuclid.org/euclid.dmj/1317149892

Digital Object Identifier
doi:10.1215/00127094-1443478

Mathematical Reviews number (MathSciNet)
MR2838352

Zentralblatt MATH identifier
1267.46072

#### 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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