## Introduction to Operator Space TheoryThe theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C* algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of "length" of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer. |

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### Contents

II | 17 |

III | 28 |

V | 34 |

VI | 40 |

VII | 42 |

IX | 43 |

X | 47 |

XI | 51 |

XLII | 178 |

XLIII | 182 |

XLIV | 183 |

XLV | 191 |

XLVI | 200 |

XLVII | 210 |

XLVIII | 215 |

XLIX | 217 |

XII | 52 |

XIII | 59 |

XIV | 63 |

XV | 64 |

XVI | 65 |

XVII | 67 |

XVIII | 68 |

XIX | 71 |

XX | 81 |

XXI | 86 |

XXIII | 92 |

XXIV | 93 |

XXVI | 98 |

XXVII | 101 |

XXVIII | 102 |

XXIX | 106 |

XXX | 109 |

XXXI | 122 |

XXXIII | 130 |

XXXIV | 135 |

XXXV | 138 |

XXXVI | 148 |

XXXVII | 165 |

XXXIX | 172 |

XL | 173 |

XLI | 175 |

### Common terms and phrases

actually admits algebra analog applied arbitrary argument associated assume Banach space bounded C*-algebra Chapter claim clearly closed commuting completely contractive completely isometric completely isomorphic conclude Consider constant converse Corollary corresponding defined definition denote direct element equal equipped equivalent exact example Exercise extends fact factorization finite finite-dimensional function given hence Hilbert space holds homomorphism identified identity implies inclusion inequality injective Lemma linear map matrix maximal Mn(E Moreover morphism natural Neumann algebra norm Note nuclear observe obtain obvious Omin operator algebra operator space operator space structure particular positive possible preceding projection Proof Proposition proved quotient Recall reflexive Remark representation resp respectively restriction result runs satisfies sense separable sequence subspace suffices taking tensor product Theorem theory unital unitary von Neumann

### Popular passages

Page 471 - V. Peller, Estimates of functions of power bounded operators on Hilbert space, J. Operator Theory 7 (1982), 341-372.

Page 469 - E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.

Page 471 - ¡M 05.0400 47A65 (46L05) Completely bounded homomorphisms of operator algebras. Proc. Amer. Math. Soc. 92 (1984), no. 2, 225-228. (GA Elliott) 8Sm:47049 46L05.0380 47D25 (46L05) (with Suen, Ching Yun) Commutant representations of completely bounded maps.

Page 471 - G. Pisier. A simple proof of a theorem of Jean Bourgain. Michigan Math. J. 39 (1992), 475—484.