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
7 - Arveson's Extension Theorems
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 this chapter we extend the results of Chapter 6 from finite-dimensional ranges Mn to maps with range B(H). We then develop the immediate applications of the extension theorems to dilation theory. We begin with some observations of a general functional-analytic nature.
Let X and Y be Banach spaces, let Y* denote the dual of Y, and let B(X, Y*) denote the bounded linear transformations of X into Y*. We wish to construct a Banach space such that B(X, Y*) is isometrically isomorphic to its dual. This will allow us to endow B(X, Y*) with a weak* topology.
Fix vectors x in X and y in Y, and define a linear functional x ⊗ y on B(X, Y*) by x ⊗ y(L) = L(x)(y). Since |x ⊗ y(L)| ≤ ||L||·||x||·||y||, we see that x ⊗ y is in B(X, Y*)* with ||x ⊗ y|| ≤ ||x|| ||y||. In fact, ||x ⊗ y|| = ||x|| ||y|| (Exercise 7.1).
It is not difficult to check that the above definition is bilinear, i.e., x ⊗ (y1 + y2) = x ⊗ y1 + x ⊗ y2, (x1 + x2) ⊗ y = x1 ⊗ y + x2 ⊗ y, and (λx) ⊗ y = x ⊗ (λy) = λ(x ⊗ y) for λ ∈. We let Z denote the closed linear span in B(X, Y*)* of these elementary tensors. Actually, Z can be identified as the completion of X ⊗ Y with respect to a cross-norm (Exercise 7.1), but we shall not need that fact here.
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- Completely Bounded Maps and Operator Algebras , pp. 84 - 96Publisher: Cambridge University PressPrint publication year: 2003