## Operator Algebras and Their Modules: An Operator Space ApproachThis invaluable reference is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area. A major trend in modern mathematics, inspired largely by physics, is toward `noncommutative' or `quantized' phenomena. In functional analysis, this has appeared notably under the name of `operator spaces', which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in 'noncommutative mathematics'. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, Non-selfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important non-commutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a lengthy section of notes containing a wealth of additional information. |

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### Contents

1 Operatorspaces | 1 |

2 Basic theory of operator algebras | 49 |

3 Basic theory of operator modules | 102 |

4 Some extremal theory | 147 |

5 Completely isomorphic theory of operator algebras | 195 |

6 Tensor products of operator algebras | 232 |

7 Selfadjointness criteria | 269 |

8 Cmodules and operator spaces | 296 |

Appendix | 359 |

369 | |

385 | |

### Common terms and phrases

A-module map adjoint approximately unital operator assume Banach algebra Banach space bilinear bimodule C∗-algebra C∗-module CB(X commutative completely contractive homomorphism completely contractive map completely contractive representation completely isometrically isomorphic completely isomorphic completely positive contractive map Conversely Corollary deduce define denote diagonal direct sum dual operator algebra dual operator space et)t example exists extends fact follows Haagerup tensor product Hence Hilbert modules Hilbert space idempotent identity map injective envelope inner product Lemma linear map LM(A M-ideals matrix normed Mn(A Mn(X module map Morita equivalent morphism multiplier multiplier algebra noncommutative nondegenerate operator modules operator space structure predual projection proof Proposition prove Q-algebra quotient map resp result right C*-module selfadjoint separately w*-continuous Shilov boundary shows space H subalgebra submodule subspace Suppose surjective Theorem theory uniform algebra unital C∗-algebra unital completely contractive unital operator algebra universal property w*-topology write