Pisier's operator Hilbert space


Richard M. Timoney


Analysis Seminar 25/2/1997


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



David P. Blecher, The standard dual of an operator space, Pacific J. Math., 153, 15-30, 1992.
David P. Blecher, Vern I. Paulsen, Tensor products of operator spaces, J. Funct. Anal., 99, 262-292, 1991.
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.
E. G. Effros, Z.-J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc., 119, 579-584, 1993.
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.
Vern I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Math. 146, Longman, 1986.
Gilles Pisier, The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms, Memoirs number 585, Amer. Math. Soc., July, 1996.
Z.-J. Ruan, Subspaces of C*-algebras, J. Funct. Anal., 76, 217-230, 1988.