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
17 - Completely Bounded Multilinear Maps and the Haagerup Tensor Norm
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 many parts of analysis an important role is played by multilinear maps. Recall that if E, F, and Z are vector spaces, then a map γ: E × F → Z is bilinear provided that it is linear in each variable, i.e., γ(e1 + e2, f1) = γ(e1, f1) + γ(e2, f1), γ(e1, f1 + f2) = γ(e1, f1) + γ(e1, f2), and γ(λe1, f1) = γ(e1, λf1) = λγ(e1, f1) for any e1 and e2 in E, for any f1 and f2 in F, and for any λ in. If one forms the algebraic tensor product E ⊗ F of E and F, then there is a one-to-one correspondence between linear maps Г: E ⊗ F → Z and bilinear maps γ: E × F → Z given by setting Г(e ⊗ f) = γ(e, f).
Consequently, if one endows E ⊗ F with a matrix norm, then the completely bounded linear maps from E ⊗ F to another matrix-normed space Z correspond to a family of bilinear maps from E × F to Z that one would like to regard as the “completely bounded” bilinear maps. In this fashion, one often arrives at an important family of bilinear maps to study.
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- Chapter
- Information
- Completely Bounded Maps and Operator Algebras , pp. 239 - 259Publisher: Cambridge University PressPrint publication year: 2003