# Pisier's operator Hilbert space

## Author

Richard M. Timoney## Date

Analysis Seminar 25/2/1997# Abstract

The object of this seminar is to explain the definition of Pisier's operator
Hilbert space, as is done in the beginning of his monograph
[7] . Pisier's definition and proof of the main universal properties take
only the first few pages of the monograph. However, one needs to be
comfortable with thinking in terms of certain of the concepts in the theory of
operator spaces before one can digest these few pages. We begin with an
overview of these aspects of the theory.

# Table of Contents

# 1 Operator Spaces

## 1.1 Concrete operator spaces

## 1.2 Abstract operator spaces

# 2 Examples of operator spaces

# 3 The dual of an operator space is an operator
space

# 4 Examples of duals of operator spaces

## 4.1 The complex conjugate of a space

# 5 The operator Hilbert Space

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# References

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- David P. Blecher,
*The standard dual of an
operator space*, Pacific J. Math., **153**, 15-30, 1992.
- [2]
- David P. Blecher, Vern I. Paulsen,
*Tensor
products of operator spaces*, J. Funct. Anal., **99**, 262-292, 1991.
- [3]
- Jean de Canniere, Uffe
Haagerup,
*Multipliers of the Fourier algebras of some simple Lie groups and
their discrete subgroups*, Amer. J. Math, **107**, 455-500, 1985.
- [4]
- E. G. Effros, Z.-J. Ruan,
*On the abstract
characterization of operator spaces*, Proc. Amer. Math. Soc., **119**,
579-584, 1993.
- [5]
- U. Haagerup,
*Injectivity and decomposition
of completely bounded maps*, Operator Algebras and their Connection with
Topology and Ergodic Theory, 170-222, Springer Lecture Notes in Math. 1132,
1985.
- [6]
- Vern I. Paulsen, Completely Bounded Maps and
Dilations,
**Pitman Research Notes in Math. 146**, Longman, 1986.
- [7]
- Gilles Pisier, The Operator Hilbert Space OH,
Complex Interpolation and Tensor Norms,
**Memoirs number 585**, Amer. Math.
Soc., July, 1996.
- [8]
- Z.-J. Ruan,
*Subspaces of C*-algebras*,
J. Funct. Anal., **76**, 217-230, 1988.