Let $ 1< p < \infty$. It is shown that if $G$ is a discrete group
with the approximation property introduced by U. Haagerup and
J. Kraus, then the noncommutative $L\sb p(VN(G))$-space has the
operator space approximation property. If, in addition, the group von
Neumann algebra $VN(G)$ has the quotient weak expectation property
(QWEP), that is, is a quotient of a $C\sp \ast$-algebra with Lance's
weak expectation property, then $L\sb p(V N(G))$ actually has the
completely contractive approximation property and the approximation
maps can be chosen to be finite-rank completely contractive
multipliers on $L\sb p(V N(G))$. Finally, we show that if $G$ is a
countable discrete group having the approximation property and $V
N(G)$ has the QWEP, then $L\sb p(V N(G))$ has a very nice local
structure; that is, it is a $\mathscr {COL}\sb p$-space and has a
completely bounded Schauder basis.