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Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’

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Abstract

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 :

$$\frac{1}{108}\sqrt{\frac{n}{1+\ln{n}}} \le \inf_{P:\mathcal{B}(\ell_2)\to{OH}_n, P^2=P} \left\|P\right\|_{cb} \le 288\pi\sqrt{\frac{2n}{1+\ln{n}}}.$$

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 OH n 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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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Marius Junge

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Mathematics Subject Classification (2000)

47L25

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Junge, M. Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’. Invent. math. 161, 225–286 (2005). https://doi.org/10.1007/s00222-004-0421-0

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