## Completely Bounded Maps and Operator AlgebrasThis book tours the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, along with some of their main applications. It requires only a basic background in functional analysis. The presentation is self-contained and paced appropriately for graduate students new to the subject. Experts will appreciate how 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 indispensible introduction to the theory of operator spaces. |

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### 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

abstract apply assume Banach space C*-algebra Chapter characterization Clearly closed compact completely bounded maps completely contractive completely isometric completely positive maps condition consequently consider constant contained continuous Conversely Corollary corresponding decomposition define denote dense dilation element entries equal equivalent example Exercise exists extends fact factorization finite give given Haagerup Hence Hilbert space homomorphism identify identity implies induced inequality introduced isometrically isomorphic known Lemma linear map matrix measure minimal multiplication norm normal Note obtain operator algebra operator space operator system pair polynomially bounded positive definite Proof Proposition Prove range Recall regard representation respectively result satisfying scalar sequence Show similar space H spectral set structure subalgebra subspace supremum tensor product Theorem theory unique unital unital C*-algebra unital homomorphism unitary verify yields