Embedding of the operator space OH and the logarithmic `little Grothendieck inequality'

Using free random varaibles we find an embedding of the operator space $OH$ in the predual of a von Neumann algebra. The properties of this embedding allow us to determined the projection constant of $O{H}_n$, i.e. there exists a projection $P:B({\mathrm{\ell }}_2)\to O{H}_n$ whose completely bounded norm behaves as n^{1/2}/(1+ln n)^{1/2}. According to recent results of Pisier/Shlyahtenko, the lower bound holds for every projection. Improving a previous estimate of order $(1+lnn)$ of the author, Pisier/Shlyahtenko obtained a logarithmic little Grothendieck inequality'. We find a second proof of this inequality which explains why the factor $\sqrt{1+\ln n}$ is indeed necessary. In particular the operator space version of the little Grothendieck inequality’ fails to hold. This `logarithmic little Grothendieck’ inequality characterizes $C^{\ast }$-algebras with the weak expectation property of Lance.