Foundations of Free Noncommutative Function Theory
Seiten
2014
American Mathematical Society (Verlag)
978-1-4704-1697-3 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-1697-3 (ISBN)
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Develops 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 to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include noncommutative polynomials, power series, and rational expressions.
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, quantum control.
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, quantum control.
Dmitry S. Kaliuzhnyi-Verbovetskyi, Drexel University, Philadelphia, PA, USA. Victor Vinnikov, Ben Gurion University of the Negev, Beer Sheva, Israel.
Introduction
NC functions and their difference-differential calculus
Higher order nc functions and their difference-differential calculus
The Taylor-Taylor formula
NC functions on nilpotent matrices
NC polynomials vs. polynomials in matrix entries
NC analyticity and convergence of TT series
Convergence of nc power series
Direct summands extensions of nc sets and nc functions (Some) earlier work on nc functions
Similarity invariant envelopes and extension of nc functions
Bibliography
Index
| Erscheint lt. Verlag | 30.12.2014 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 456 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-1697-2 / 1470416972 |
| ISBN-13 | 978-1-4704-1697-3 / 9781470416973 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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