  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 $OH_n$, i.e. there exists a
projection $P:B(\ell_2)\to OH_n$ 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
$(1+ ln n)$ 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 $C^*$-algebras
with the weak expectation property of Lance.
