  Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the
approximation property introduced by Haagerup and Kraus, then the
non-commutative $L_p(VN(G))$ space has the operator space approximation
property. If, in addition, the group von Neumann algebra $VN(G)$ has the QWEP,
i.e. is a quotient of a $C^*$-algebra with Lance's weak expectation property,
then $L_p(VN(G))$ actually has the completely contractive approximation
property and the approximation maps can be chosen to be finite-rank completely
contractive multipliers on $L_p(VN(G))$. Finally, we show that if $G$ is a
countable discrete group having the approximation property and $VN(G)$ has the
QWEP, then $L_p(VN(G))$ has a very nice local structure, i.e. it is a $\mathcal
C\OL_p$ space and has a completely bounded Schauder basis.
