Approximation properties for noncommutative Lp-spaces associated with discrete groups

Abstract

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.