### Abstract

Following Grothendieck's characterization of Hilbert spaces we consider operator spaces F such that both F and F^{*} completely embed into the dual of a C*-algebra. Due to Haagerup/Musat's improved version of Pisier/Shlyakhtenko's Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum C⊕R of the column and row spaces (the corresponding class being denoted by QS(C⊕R)). We first prove a representation theorem for homogeneous F∈QS(C⊕R) starting from the fundamental sequences given by an orthonormal basis (e_{k}) of F. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of F and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an n-dimensional subspace F_{n} of F: In the same way, the projection (=injectivity) constant of F_{n} is explicitly expressed in terms of Φ_{c} and Φ_{r} too. Orlicz space techniques play a crucial role in our arguments. They also permit us to determine the completely 1-summing maps in Effros and Ruan's sense between two homogeneous spaces E and F in QS(C⊕R). The resulting space Π_{1}^{o}(E, F) isomorphically coincides with a Schatten-Orlicz class S_{φ}. Moreover, the underlying Orlicz function φ is uniquely determined by the fundamental sequences of E and F. In particular, applying these results to the column subspace C_{p} of the Schatten p-class, we find the projection and exactness constants of C_{p}^{n}, and determine the completely 1-summing maps from C_{p} to C_{q} for any 1≤p, q≤∞.

Original language | English (US) |
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Pages (from-to) | 75-118 |

Number of pages | 44 |

Journal | Inventiones Mathematicae |

Volume | 179 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Inventiones Mathematicae*,

*179*(1), 75-118. https://doi.org/10.1007/s00222-009-0210-x