Mathematical surveys and monographs· 2014· Book chapter
Foundations of Free Noncommutative Function Theory
- 189citations
- 2014year
Short summary
This book introduces a new theory of free noncommutative functions, defined as matrix-valued mappings that preserve matrix size, direct sums, and similarities.
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Key points
- Introduces free noncommutative functions as matrix-valued mappings.
- These functions preserve matrix size, direct sums, and similarities.
- Examples include noncommutative polynomials, power series, and rational expressions.
- Motivated by free probability and applicable to "dimensionless" matrix inequalities in system engineering.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Mathematical surveys and monographs, 2014 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics