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
3 - Completely Positive Maps
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
Let A be a C*-algebra, and let be a subspace. Then we shall call an operator space. Clearly, Mn can be regarded as a subspace of Mn(A), and we let Mn have the norm structure that it inherits from the (unique) norm structure on the C*-algebra Mn(A). We make no attempt at this time to define a norm structure on Mn without reference to A. Thus, one thing that distinguishes from an ordinary normed space is that it comes naturally equipped with norms on Mn for all n ≥ 1. Later in this book we shall give a more axiomatic definition of operator spaces, at which time we shall begin to refer to subspaces of C*-algebras as concrete operator spaces. For now we simply stress that by an operator space we mean a concrete subspace of a C*-algebra, together with this extra “baggage” of a well-defined sequence of norms on Mn. Similarly, if S ⊆ A is an operator system, then we endow Mn(S) with the norm and order structure that it inherits as a subspace of Mn(A).
As before, if B is a C*-algebra and ϕ: S → B is a linear map, then we define ϕn: Mn(S) → Mn(B) by ϕn((ai, j)) = (ϕ(ai, j)). We call ϕ n-positive if ϕn is positive, and we call ϕ completely positive if ϕ is n-positive for all n.
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- Completely Bounded Maps and Operator Algebras , pp. 26 - 42Publisher: Cambridge University PressPrint publication year: 2003