Approximation Properties for Non-commutative L_p-Spaces Associated with Discrete Groups

M. Junge; Z. -J. Ruan

paper

Approximation Properties for Non-commutative L_p-Spaces Associated with Discrete Groups

arXiv:math/0206060

Abstract

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 $L_{p} (V N (G))$ space has the operator space approximation property. If, in addition, the group von Neumann algebra $V N (G)$ has the QWEP, i.e. 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, i.e. it is a $\mathcal C\OL_p$ space and has a completely bounded Schauder basis.