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

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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.

Original languageEnglish (US)
Pages (from-to)225-286
Number of pages62
JournalInventiones Mathematicae
Issue number2
StatePublished - Aug 2005

ASJC Scopus subject areas

  • Mathematics(all)


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