# The Operator Hilbert Space \(OH\), Complex Interpolation and Tensor Norms

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*Gilles Pisier*

In the recently developed duality theory of operator spaces (as
developed by Effros-Ruan and Blecher-Paulsen) bounded operators are
replaced by completely bounded ones, isomorphisms
by complete isomorphisms, and Banach spaces by
operator spaces. This allows for distinguishing between the
various ways in which a given Banach space can be embedded isometrically
into \(B(H)\) (with \(H\) being Hilbert). In this
new category, several operator spaces which are isomorphic (as
Banach spaces) to a Hilbert space play an important role. For instance
the row and column Hilbert spaces and several other examples
appearing naturally in the construction of the Boson or Fermion Fock
spaces have been studied extensively.

One of the main results of this memoir is the observation that there
is a central object in this class: there is a unique self
dual Hilbertian operator space (denoted by \(OH\) ) which seems
to play the same central role in the category of operator spaces
that Hilbert spaces play in the category of Banach spaces.

This new concept, called “the operator Hilbert space” and
denoted by \(OH\), is introduced and thoroughly studied in this
volume.

#### Table of Contents

# Table of Contents

## The Operator Hilbert Space $OH$, Complex Interpolation and Tensor Norms

- Table of Contents vii8 free
- Introduction 110 free
- §1. The operator Hilbert space 1019 free
- §2. Complex interpolation 2130
- §3. The oh tensor product 3948
- §4. Weights on partially ordered vector spaces 5059
- §5. (2,w)-summing operators 5766
- §6. The gamma-norms and their dual norms 6776
- §7. Operators factoring through OH 7584
- §8. Factorization through a Hilbertian operator space 8089
- §9. On the "local theory" of operator spaces 8897
- §10. Open questions 95104
- References 98107