Noncommutative Lp-spaces without the completely bounded approximation property

Abstract

For any $1\le p\le \infty$ 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 $p>4$ or $p<4/3$ and $r\ge 3$ these examples are the noncommutative $L^p$-spaces of the von Neumann algebra of lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ or in $\mathrm{S}{\mathrm{L}}_r\left(\mathbb{R}\right)$. For other values of $p$ the examples are the noncommutative $L^p$-spaces of the von Neumann algebra of lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ for $r$ large enough depending on $p$.

We also prove that if $r\ge 3$ lattices in $\mathrm{S}{\mathrm{L}}_r\left(F\right)$ or $\mathrm{S}{\mathrm{L}}_r\left(\mathbb{R}\right)$ do not have the approximation property of Haagerup and Kraus. This provides examples of exact $C^{\ast }$-algebras without the operator space approximation property.