Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Positive Maps
- 3 Completely Positive Maps
- 4 Dilation Theorems
- 5 Commuting Contractions on Hilbert Space
- 6 Completely Positive Maps into Mn
- 7 Arveson's Extension Theorems
- 8 Completely Bounded Maps
- 9 Completely Bounded Homomorphisms
- 10 Polynomially Bounded and Power-Bounded Operators
- 11 Applications to K-Spectral Sets
- 12 Tensor Products and Joint Spectral Sets
- 13 Abstract Characterizations of Operator Systems and Operator Spaces
- 14 An Operator Space Bestiary
- 15 Injective Envelopes
- 16 Abstract Operator Algebras
- 17 Completely Bounded Multilinear Maps and the Haagerup Tensor Norm
- 18 Universal Operator Algebras and Factorization
- 19 Similarity and Factorization
- Bibliography
- Index
14 - An Operator Space Bestiary
Published online by Cambridge University Press: 24 November 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Positive Maps
- 3 Completely Positive Maps
- 4 Dilation Theorems
- 5 Commuting Contractions on Hilbert Space
- 6 Completely Positive Maps into Mn
- 7 Arveson's Extension Theorems
- 8 Completely Bounded Maps
- 9 Completely Bounded Homomorphisms
- 10 Polynomially Bounded and Power-Bounded Operators
- 11 Applications to K-Spectral Sets
- 12 Tensor Products and Joint Spectral Sets
- 13 Abstract Characterizations of Operator Systems and Operator Spaces
- 14 An Operator Space Bestiary
- 15 Injective Envelopes
- 16 Abstract Operator Algebras
- 17 Completely Bounded Multilinear Maps and the Haagerup Tensor Norm
- 18 Universal Operator Algebras and Factorization
- 19 Similarity and Factorization
- Bibliography
- Index
Summary
In the last chapter we obtained an abstract characterization of operator spaces that allows us to define these spaces without a concrete representation. This result has had a tremendous impact and has led to the development of a general theory of operator spaces that parallels in some ways the development of the theory of Banach spaces.
In this chapter we give the reader a brief introduction to some of the basics of this theory, focusing on some of the more important operator spaces that we will encounter in later chapters.
We have already encountered one example of the power of this axiomatic characterization. In Exercise 13.3, it was shown that if V is an operator space and W ⊆ V a closed subspace, then V / W is an operator space, where the matrix norm structure on V/W comes from the identification Mm,n(V/W) = Mm,n(V)/Mm,n(W). Yet in most concrete situations it is difficult to actually exhibit a concrete completely isometric representation of V/W as operators on a Hilbert space.
The first natural question in the area is as follows: If V is, initially, just a normed space, then is it always possible to assign norms ||·||m,n to Mm,n(V) for all m and n in such a fashion that V becomes an operator space? The answer to this question is yes.
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- Information
- Completely Bounded Maps and Operator Algebras , pp. 186 - 205Publisher: Cambridge University PressPrint publication year: 2003