Inventiones mathematicae

, Volume 161, Issue 2, pp 225–286 | Cite as

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 OHn:
$$\frac{1}{108}\sqrt{\frac{n}{1+\ln{n}}} \le \inf_{P:\mathcal{B}(\ell_2)\to{OH}_n, P^2=P} \left\|P\right\|_{cb} \le 288\pi\sqrt{\frac{2n}{1+\ln{n}}}.$$
The lower estimate is a recent result of Pisier and Shlyakhtenko that improves an estimate of order 1/(1+lnn) of the author. The additional factor \(1 / \sqrt{1+\ln{n}}\) indicates that the operator space OHn behaves differently than its classical counterpart \(\ell_2^n\). We give an application of this formula to positive sesquilinear forms on \(\mathcal{B}(\ell_2)\). This leads to logarithmic characterization of C*-algebras with the weak expectation property introduced by Lance.


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© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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