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The 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.

Operator spaces. --- Matricially normed spaces --- Non-commutative Banach space theory --- Banach spaces --- Operator theory --- Operator spaces

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Functional analysis --- 51 <082.1> --- Mathematics--Series --- Operator algebras. --- Operator spaces. --- Operator ideals. --- Algèbres d'opérateurs --- Espaces d'opérateurs --- Idéaux d'opérateurs --- Operator algebras --- Operator ideals --- Operator spaces --- Matricially normed spaces --- Non-commutative Banach space theory --- Banach spaces --- Operator theory --- Ideals (Algebra) --- Algebras, Operator --- Topological algebras --- Algèbres d'opérateurs. --- Espaces d'opérateurs. --- Idéaux d'opérateurs.

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Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include: (1) Embeddability of locally finite metric spaces into Banach spaces is finitely determined; (2) Constructions of embeddings; (3) Distortion in terms of Poincaré inequalities; (4) Constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees; (5) Banach spaces which do not admit coarse embeddings of expanders; (6) Structure of metric spaces which are not coarsely embeddable into a Hilbert space; (7) Applications of Markov chains to embeddability problems; (8) Metric characterizations of properties of Banach spaces; (9) Lipschitz free spaces. Substantial part of the book is devoted to a detailed presentation of relevant results of Banach space theory and graph theory. The final chapter contains a list of open problems. Extensive bibliography is also included. Each chapter, except the open problems chapter, contains exercises and a notes and remarks section containing references, discussion of related results, and suggestions for further reading. The book will help readers to enter and to work in a very rapidly developing area having many important connections with different parts of mathematics and computer science.

Banach spaces. --- Lipschitz spaces. --- Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Hölder spaces --- Function spaces --- Functions of complex variables --- Generalized spaces --- Topology --- Banach Space Theory. --- Bilipschitz Embedding. --- Coarse Embedding. --- Embedding of Discrete Metric Spaces. --- Functional Analysis. --- Graph Theory.