Foundations of Free Noncommutative Function Theory
About this Title
Dmitry S. Kaliuzhnyi-Verbovetskyi, Drexel University, Philadelphia, PA and Victor Vinnikov, Ben Gurion University of the Negev, Beer Sheva, Israel
Publication: Mathematical Surveys and Monographs
Publication Year: 2014; Volume 199
ISBNs: 978-1-4704-1697-3 (print); 978-1-4704-2001-7 (online)
MathSciNet review: MR3244229
MSC: Primary 46L52; Secondary 16S10, 47A60, 81Q12, 81R05, 81R10, 81R60, 81T05
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.
Table of Contents
- Chapter 1. Introduction
- Chapter 2. NC functions and their difference-differential calculus
- Chapter 3. Higher order nc functions and their difference-differential calculus
- Chapter 4. The Taylor-Taylor 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