Approximation properties for noncommutative L_p-spaces associated with discrete groups

Let 1 < p < ∞. 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_p(V N(G))-space has the operator space approximation property. If, in addition, the group von Neumann algebra V N(G) has the quotient weak expectation property (QWEP), that is, is a quotient of a C*-algebra with Lance's weak expectation property, then L_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_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_P(V N(G)) has a very nice local structure; that is, it is a script C sign script O sign ℒ_p-space and has a completely bounded Schauder basis.