Operator spaces.

*(English)*Zbl 0969.46002
London Mathematical Society Monographs. New Series. 23. Oxford: Clarendon Press. xvi, 363 p. (2000).

The book is written by two of the major contributors to the operator space theory. It is very good modern introduction to the subject and is very well suited for an introductory course. Most of basic notions are treated in an elegant way and the proofs of the main results are accessible to a non-expert.

The book is divided into 5 unequal parts. The biggest Part I is devoted to the three fundamental results upon which operator space theory is based. The first one is the representation theorem of Ruan showing that an abstract operator space can be realized as a linear space of bounded operators on a Hilbert space \(H\). The second one is the Arveson-Wittstock generalization of the Hahn-Banach theorem, which asserts that complete contractions into \(B(H)\) extend from subspaces to operator spaces. The third one is the Paulsen-Wittstock decomposition theorem for complete contractions. Part I also contains standard constructions of operator spaces (mapping spaces, quantizations, Hilbert operator spaces etc.) and a short chapter on injective operator spaces.

Part II deals with operator space tensor products. Besides the usual projective and injective tensor products, it includes the Haagerup tensor product, infinite matrices over an operator space and ultraproducts. Part III starts with the operator space analog of the Grothendieck approximation property, which is followed by the treatment of completely nuclear mappings, completely integral mappings, absolutely summing mappings and the operator space version of the Dvoretzky-Rogers theorem on unconditional summability.

Part IV studies the three key notions of the \(C^*\)-algebra theory: nuclearity, exactness and local reflexivity. Relations between these notions are discussed and the local reflexivity property for \(C^*\)-algebraic duals is established. The last and the shortest Part V contains an application to non-commutative harmonic analysis and a non-selfadjoint version of the Gelfand-Naimark theorem, which gives an abstract characterization of unital non-selfadjoint operator algebras.

The book is divided into 5 unequal parts. The biggest Part I is devoted to the three fundamental results upon which operator space theory is based. The first one is the representation theorem of Ruan showing that an abstract operator space can be realized as a linear space of bounded operators on a Hilbert space \(H\). The second one is the Arveson-Wittstock generalization of the Hahn-Banach theorem, which asserts that complete contractions into \(B(H)\) extend from subspaces to operator spaces. The third one is the Paulsen-Wittstock decomposition theorem for complete contractions. Part I also contains standard constructions of operator spaces (mapping spaces, quantizations, Hilbert operator spaces etc.) and a short chapter on injective operator spaces.

Part II deals with operator space tensor products. Besides the usual projective and injective tensor products, it includes the Haagerup tensor product, infinite matrices over an operator space and ultraproducts. Part III starts with the operator space analog of the Grothendieck approximation property, which is followed by the treatment of completely nuclear mappings, completely integral mappings, absolutely summing mappings and the operator space version of the Dvoretzky-Rogers theorem on unconditional summability.

Part IV studies the three key notions of the \(C^*\)-algebra theory: nuclearity, exactness and local reflexivity. Relations between these notions are discussed and the local reflexivity property for \(C^*\)-algebraic duals is established. The last and the shortest Part V contains an application to non-commutative harmonic analysis and a non-selfadjoint version of the Gelfand-Naimark theorem, which gives an abstract characterization of unital non-selfadjoint operator algebras.

Reviewer: V.M.Manuilov (Moskva)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L07 | Operator spaces and completely bounded maps |

47L25 | Operator spaces (= matricially normed spaces) |

46M05 | Tensor products in functional analysis |

46B28 | Spaces of operators; tensor products; approximation properties |