**University Lecture Series**

Volume: 56;
2010;
241 pp;
Softcover

MSC: Primary 46; 47;

Print ISBN: 978-0-8218-5254-5

Product Code: ULECT/56

List Price: $54.00

AMS Member Price: $43.20

MAA member Price: $48.60

**Electronic ISBN: 978-1-4704-1651-5
Product Code: ULECT/56.E**

List Price: $54.00

AMS Member Price: $43.20

MAA member Price: $48.60

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#### Supplemental Materials

# Quantum Functional Analysis: Non-Coordinate Approach

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*A. Ya. Helemskii*

This book contains a systematic presentation of quantum functional
analysis, a mathematical subject also known as operator space theory.
Created in the 1980s, it nowadays is one of the most prominent areas
of functional analysis, both as a field of active research and as a
source of numerous important applications.

The approach taken in this book differs significantly from the
standard approach used in studying operator space theory. Instead of
viewing “quantized coefficients” as matrices in a fixed
basis, in this book they are interpreted as finite rank operators in a
fixed Hilbert space. This allows the author to replace matrix
computations with algebraic techniques of module theory and tensor
products, thus achieving a more invariant approach to the subject.

The book can be used by graduate students and research
mathematicians interested in functional analysis and related areas of
mathematics and mathematical physics. Prerequisites include standard
courses in abstract algebra and functional analysis.

#### Readership

Graduate students and research mathematicians interested in functional analysis.

#### Reviews & Endorsements

This book is highly recommended to mathematicians who wish to become acquainted with the research field of operator spaces, operator modules and operator algebras.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Quantum Functional Analysis: Non-Coordinate Approach

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Introduction vii8 free
- Three basic definitions and three principal theorems 120 free
- Part I. The beginning: Spaces and operators 726
- Preparing the stage 928
- Abstract operator ( = quantum) spaces 3352
- Completely bounded operators 4766
- The completion of abstract operator spaces 6382
- Part II. Bilinear operators, tensor products and duality 6786
- Strongly and weakly completely bounded bilinear operators 6988
- New preparations: Classical tensor products 7594
- Quantum tensor products 81100
- Quantum duality 119138
- Part III. Principal theorems, revisited in earnest 151170
- Extreme flatness and the extension theorem 153172
- Representation theorem and its gifts 167186
- Decomposition theorem 177196
- Returning to the Haagerup tensor product 189208
- Miscellany: More examples, facts and applications 201220
- Bibliography 231250
- Index 237256 free
- Back Cover Back Cover1264