## 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 A | 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 ASpectral Sets | 150 |

285 | |

297 | |

### Common terms and phrases

assume bilinear bimodule bounded homomorphism bounded linear bounded linear map bounded operator C-bimodule map C*-subalgebra Chapter commuting contractions compact completely bounded maps completely contractive completely isometric isomorphism completely positive definite completely positive maps contractive homomorphism Corollary decomposition define denote dilation theorem example Exercise exists a Hilbert extends extension theorem factorization finite g>max given Hausdorff space Hence Hilbert space I(Sx identify identity map implies infimum injective envelope integers Lemma linear functional linear map Math matrix norm minimal Mn(A Mn(S n-tuple Neumann's inequality normed space Note obtain operator algebra operator space structure operator system operator-valued Pisier pletely polynomially bounded positive elements power-bounded proof of Theorem Proposition Prove representation theorem result satisfying Schur product self-adjoint sequence similar space H spectral set Stinespring representation subalgebra subset subspace supremum tensor product theory unital C*-algebra unital homomorphism unital operator algebra unitary dilation vector space Wittstock