## 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}: 1/108√n/1+ln n ≤ inf _{P:B(ℓ2)→OHn}, P^{2}=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 ℓ_{2}^{n}. We give an application of this formula to positive sesquilinear forms on B(ℓ_{2}). This leads to logarithmic characterization of C*-algebras with the weak expectation property introduced by Lance.

Original language | English (US) |
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Pages (from-to) | 225-286 |

Number of pages | 62 |

Journal | Inventiones Mathematicae |

Volume | 161 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2005 |

## ASJC Scopus subject areas

- General Mathematics