Foundations of Free Noncommutative Function Theory

Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov

Research output: Book/ReportBookpeer-review


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.
Original languageEnglish
Place of PublicationProvidence, Rhode Island
PublisherAmerican Mathematical Society
Number of pages183
ISBN (Electronic)9781470420017
ISBN (Print)9781470416973
StatePublished - 2014

Publication series

NameMathematical surveys and monographs
PublisherAmerican Mathematical Society
Volume 199


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