NONCOMMUTATIVE Lp-SPACES WITHOUT THE COMPLETELY BOUNDED APPROXIMATION PROPERTY

For any 1 <= p <= infinity different from 2, we give examples of noncommutative L-p-space's without the completely bounded approximation property. Let F be a nonarchimedian local field. If p > 4 or p < 4/3 and r >= 3 these examples are the noncommutative L-p-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 L-p-spaces of the von Neumann algebra of lattices in SLr (F) for r large enough depending on p. We also prove that if r >= 3 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.