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

We use Voiculescu's concept of free probability to construct a completely isomorphic embedding of the operator space OH in the predual of a von Neumann algebra. We analyze the properties of this embedding and determine the operator space projection constant of OH _n: 1/108√n/1+ln n ≤ inf _{P:B(ℓ2)→OHn}, P²=P ∥P∥_cb≤ 288π√2n/1+ln n. The lower estimate is a recent result of Pisier and Shlyakhtenko that improves an estimate of order 1/(1+ln n) of the author. The additional factor 1/√1+ln n indicates that the operator space OH _n behaves differently than its classical counterpart ℓ₂ⁿ. We give an application of this formula to positive sesquilinear forms on B(ℓ₂). This leads to logarithmic characterization of C*-algebras with the weak expectation property introduced by Lance.