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
1 - Introduction
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
It is assumed throughout this book that the reader is familiar with operator theory and the basic properties of C*-algebras (see for example [76] and [8, Chapter 1]). We concentrate primarily on giving a self-contained exposition of the theory of completely positive and completely bounded maps between C*-algebras and the applications of these maps to the study of operator algebras, similarity questions, and dilation theory. In particular, we assume that the reader is familiar with the material necessary for the Gelfand–Naimark–Segal theorem, which states that every C*-algebra has a one-to-one, ∗-preserving, norm-preserving representation as a norm-closed, ∗-closed algebra of operators on a Hilbert space.
In this chapter we introduce some of the key concepts that will be studied in this book.
As well as having a norm, a C*-algebra also has an order structure, induced by the cone of positive elements. Recall that an element of a C*-algebra is positive if and only if it is self-adjoint and its spectrum is contained in the nonnegative reals, or equivalently, if it is of the form a*a for some element a. Since the property of being positive is preserved by ∗-isomorphism, if a C*-algebra is represented as an algebra of operators on a Hilbert space, then the positive elements of the C*-algebra coincide with the positive operators that are contained in the representation of the algebra.
- Type
- Chapter
- Information
- Completely Bounded Maps and Operator Algebras , pp. 1 - 8Publisher: Cambridge University PressPrint publication year: 2003