## 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

Positive Maps | 9 |

Completely Positive Maps | 26 |

Dilation Theorems | 43 |

Commuting Contractions on Hilbert Space | 58 |

Completely Positive Maps into M | 73 |

Arvesons Extension Theorems | 84 |

Completely Bounded Maps | 97 |

Completely Bounded Homomorphisms | 120 |

Tensor Products and Joint Spectral Sets | 159 |

Abstract Characterizations of Operator Systems | 175 |

An Operator Space Bestiary | 186 |

Injective Envelopes | 206 |

Abstract Operator Algebras | 225 |

Completely Bounded Multilinear Maps and the Haagerup | 239 |

Universal Operator Algebras and Factorization | 260 |

Similarity and Factorization | 273 |

### Common terms and phrases

&max &min ai,j Banach bilinear bimodule bounded homomorphism bounded linear bounded linear map bounded operator C*-algebra with unit C*-subalgebra Chapter commuting contractions compact completely bounded maps completely contractive completely isometric isomorphism completely positive maps contractive homomorphism contractive map Corollary decomposition define denote dilation theorem example Exercise exists a Hilbert extends extension theorem factorization finite given Hausdorff space Hence Hilbert space identity map implies infimum injective envelope k-homomorphism K-spectral Lemma linear functional linear map 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's pletely polynomially bounded positive elements power-bounded proof of Theorem Proposition Prove result satisfying 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 Wittstock xi,j