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Hardcover ISBN:  9781470416973 
Product Code:  SURV/199 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470420017 
Product Code:  SURV/199.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470416973 
eBook ISBN:  9781470420017 
Product Code:  SURV/199.B 
List Price:  $254.00$191.50 
MAA Member Price:  $228.60$172.35 
AMS Member Price:  $203.20$153.20 

Book DetailsMathematical Surveys and MonographsVolume: 199; 2014; 183 ppMSC: Primary 17; 40; 46; 47;
In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions.
Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is “dimensionless” matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.ReadershipGraduate students interested in noncommutative analysis.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. NC functions and their differencedifferential calculus

Chapter 3. Higher order nc functions and their differencedifferential calculus

Chapter 4. The TaylorTaylor formula

Chapter 5. NC functions on nilpotent matrices

Chapter 6. NC polynomials vs. polynomials in matrix entries

Chapter 7. NC analyticity and convergence of TT series

Chapter 8. Convergence of nc power series

Chapter 9. Direct summands extensions of nc sets and nc functions

Chapter 10. (Some) earlier work on nc functions

Appendix A. Similarity invariant envelopes and extension of nc functions


Additional Material

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In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions.
Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is “dimensionless” matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.
Graduate students interested in noncommutative analysis.

Chapters

Chapter 1. Introduction

Chapter 2. NC functions and their differencedifferential calculus

Chapter 3. Higher order nc functions and their differencedifferential calculus

Chapter 4. The TaylorTaylor formula

Chapter 5. NC functions on nilpotent matrices

Chapter 6. NC polynomials vs. polynomials in matrix entries

Chapter 7. NC analyticity and convergence of TT series

Chapter 8. Convergence of nc power series

Chapter 9. Direct summands extensions of nc sets and nc functions

Chapter 10. (Some) earlier work on nc functions

Appendix A. Similarity invariant envelopes and extension of nc functions