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
19 - Similarity and Factorization
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 saw how the abstract characterization of operator algebras led to a number of factorization formulas for certain universal operator algebras. However, this theory was an isometric theory. In this chapter we focus on the isomorphic theory of operator algebras and applications to similarity questions.
We present Pisier's remarkable work on similarity degree and factorization degree, and Blecher's characterization of operator algebras up to cb isomorphism.
Pisier's work shows that for an operator algebra B, every bounded homomorphism is completely bounded if and only if the type of factorization occurring in the study of MAXA(B) can be carried out with uniform control on the number of factors needed. The least such integer is the factorization degree of the algebra.
Pisier's work has a number of deep implications in the study of bounded representations of groups and in the study of Kadison's similarity conjecture. We focus primarily on Kadison's conjecture, that every bounded homomorphism of a C*-algebra into B(H) is similar to a ∗-homomorphism. Thus, we will show that Kadison's conjecture is equivalent to the existence of an integer d such that every C*-algebra has factorization degree at most d.
A pivotal role in Pisier's work is played by the universal operator algebra of an operator space.
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- Completely Bounded Maps and Operator Algebras , pp. 273 - 284Publisher: Cambridge University PressPrint publication year: 2003