David P. Blecher's Books

Operator Algebras and Their Modules

An Operator Space Approach

By: David P. Blecher,Christian Le Merdy

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Operator Algebras and Their Modules

An Operator Space Approach

By: David P. Blecher,Christian Le Merdy

This 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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Page Count

398

Publisher

Clarendon Press

Published Date

ISBN 10

0198526598

ISBN 13

9780198526599

The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

By: David P. Blecher,Vrej Zarikian

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The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

By: David P. Blecher,Vrej Zarikian

The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C*$-modules and their maps. Here we give a systematic exposition of this theory. The main part of this memoir consists of a 'calculus' for one-sided $M$-ideals and multipliers, i.e. a collection of the properties of one-sided $M$-ideals and multipliers with respect to the basic constructions met in functional analysis. This is intended to be a reference tool for 'noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.

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Page Count

102

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821838237

ISBN 13

9780821838235

Categories of Operator Modules (Morita Equivalence and Projective Modules)

Morita Equivalence and Projective Modules

By: David P. Blecher,Paul S. Muhly,Vern I. Paulsen

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Categories of Operator Modules (Morita Equivalence and Projective Modules)

Morita Equivalence and Projective Modules

By: David P. Blecher,Paul S. Muhly,Vern I. Paulsen

We employ recent advances in the theory of operator spaces, also known as quantized functional analysis, to provide a context in which one can compare categories of modules over operator algebras that are not necessarily self-adjoint. We focus our attention on the category of Hilbert modules over an operator algebra and on the category of operator modules over an operator algebra. The module operations are assumed to be completely bounded - usually, completely contractive. Wedevelop the notion of a Morita context between two operator algebras A and B. This is a system (A,B,{} {A}X {B},{} {B} Y {A},(\cdot,\cdot),[\cdot,\cdot]) consisting of the algebras, two bimodules {A}X {B and {B}Y {A} and pairings (\cdot,\cdot) and [\cdot,\cdot] that induce (complete) isomorphisms betweenthe (balanced) Haagerup tensor products, X \otimes {hB} {} Y and Y \otimes {hA} {} X, and the algebras, A and B, respectively. Thus, formally, a Morita context is the same as that which appears in pure ring theory. The subtleties of the theory lie in the interplay between the pure algebra and the operator space geometry. Our analysis leads to viable notions of projective operator modules and dual operator modules. We show that two C*-algebras are Morita equivalent in our sense if and only ifthey are C*-algebraically strong Morita equivalent, and moreover the equivalence bimodules are the same. The distinctive features of the non-self-adjoint theory are illuminated through a number of examples drawn from complex analysis and the theory of incidence algebras over topological partial orders.Finally, an appendix provides links to the literature that developed since this Memoir was accepted for publication.

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Page Count

109

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082181916X

ISBN 13

9780821819166

Book's by David P. Blecher

Operator Algebras and Their Modules

Operator Algebras and Their Modules

The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces

Categories of Operator Modules (Morita Equivalence and Projective Modules)

Categories of Operator Modules (Morita Equivalence and Projective Modules)

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