**Mathematical Surveys and Monographs**

Volume: 199;
2014;
183 pp;
Hardcover

MSC: Primary 17; 40; 46; 47;

Print ISBN: 978-1-4704-1697-3

Product Code: SURV/199

List Price: $77.00

Individual Member Price: $61.60

**Electronic ISBN: 978-1-4704-2001-7
Product Code: SURV/199.E**

List Price: $77.00

Individual Member Price: $61.60

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#### Supplemental Materials

# Foundations of Free Noncommutative Function Theory

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*Dmitry S. Kaliuzhnyi-Verbovetskyi; Victor Vinnikov*

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.

#### Table of Contents

# Table of Contents

## Foundations of Free Noncommutative Function Theory

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. NC functions and their difference-differential calculus 1522 free
- Chapter 3. Higher order nc functions and their difference-differential calculus 3340
- Chapter 4. The Taylor-Taylor formula 6168
- Chapter 5. NC functions on nilpotent matrices 6774
- Chapter 6. NC polynomials vs. polynomials in matrix entries 7784
- Chapter 7. NC analyticity and convergence of TT series 8390
- Chapter 8. Convergence of nc power series 125132
- Chapter 9. Direct summands extensions of nc sets and nc functions 155162
- Chapter 10. (Some) earlier work on nc functions 167174
- Appendix A. Similarity invariant envelopes and extension of nc functions 171178
- Bibliography 175182
- Index 181188 free
- Back Cover Back Cover1194

#### Readership

Graduate students interested in noncommutative analysis.