## Completely Bounded Maps and Operator AlgebrasIn this book, first published in 2003, the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications. The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature. An indispensable introduction to the theory of operator spaces for all who want to know more. |

### Contents

Introduction | 1 |

Positive Maps | 9 |

Completely Positive Maps | 26 |

Dilation Theorems | 43 |

Commuting Contractions on Hilbert Space | 58 |

Completely Positive Maps into Mn | 73 |

Arvesons Extension Theorems | 84 |

Completely Bounded Maps | 97 |

Tensor Products and Joint Spectral Sets | 159 |

Abstract Characterizations of Operator Systems and Operator Spaces | 175 |

An Operator Space Bestiary | 186 |

Injective Envelopes | 206 |

Abstract Operator Algebras | 225 |

Completely Bounded Multilinear Maps and the Haagerup Tensor Norm | 239 |

Universal Operator Algebras and Factorization | 260 |

Similarity and Factorization | 273 |

Completely Bounded Homomorphisms | 120 |

Polynomially Bounded and PowerBounded Operators | 135 |

Applications to KSpectral Sets | 150 |

285 | |

297 | |

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### Common terms and phrases

A₁ Banach space bilinear bimodule bounded homomorphism bounded linear bounded operator C*-algebra with unit C*-subalgebra C₁ Chapter commuting contractions compact completely bounded maps completely contractive completely isometric isomorphism completely positive maps contractive homomorphism Corollary decomposition define denote dilation theorem E₁ example Exercise exists a Hilbert extension theorem factorization finite given h₁ Hausdorff space Hence Hilbert space homomorphism I(Sx identity map implies infimum injective envelope integers K-spectral K₁ Lemma linear functional linear map Math matrix norm MIN(V minimal n-tuple Neumann's inequality normed space Note operator algebra operator space structure operator system operator-valued Pisier polynomially bounded power-bounded proof of Theorem Proposition Prove result S₁ satisfying scalar self-adjoint sequence Show similar space H spectral set Stinespring representation subalgebra subspace supremum tensor product theory unital C*-algebra unital homomorphism unital operator algebra unitary dilation V₁ vector space Wittstock