Quantum Functional Analysis: Non-Coordinate Approach
About this Title
A. Ya. Helemskii, Moscow Lomonosov State University, Moscow, Russia
Publication: University Lecture Series
Publication Year 2010: Volume 56
ISBNs: 978-0-8218-5254-5 (print); 978-1-4704-1651-5 (online)
MathSciNet review: MR2760416
MSC: Primary 46L07; Secondary 46H25, 46M05, 46N50, 47L25, 47N50
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.
Graduate students and research mathematicians interested in functional analysis.
Table of Contents
Part I. The beginning: Spaces and operators
- Chapter 1. Preparing the stage
- Chapter 2. Abstract operator ( = quantum) spaces
- Chapter 3. Completely bounded operators
- Chapter 4. The completion of abstract operator spaces
Part II. Bilinear operators, tensor products and duality
- Chapter 5. Strongly and weakly completely bounded bilinear operators
- Chapter 6. New preparations: Classical tensor products
- Chapter 7. Quantum tensor products
- Chapter 8. Quantum duality
Part III. Principal theorems, revisited in earnest
- Chapter 9. Extreme flatness and the extension theorem
- Chapter 10. Representation theorem and its gifts
- Chapter 11. Decomposition theorem
- Chapter 12. Returning to the Haagerup tensor product
- Chapter 13. Miscellany: More examples, facts and applications