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Description of  Methods of Noncommutative Analysis (Contributions to the Sociology of Language)

Provides a systematic and self-contained introduction to the basics of noncommutative analysis with interesting applications, ranging from classical to modern ones. Noncommutative analysis, that is, the calculus of noncommuting operators, is one of the main tools in contemporary mathematics. Indeed, the theory of differential and pseudodifferential operators, various problems of algebra, functional analysis, and theoretical physics deal with functions of noncommuting operators. Noncommutative analysis occurs to be useful in such mathematical fields as the theory of geometric and asymptotic quantization, representation theory, the theory of quantum groups, and so on. In this book the reader will find a lot of examples from functional analysis, algebra, representation theory, and the theory of differential equations where non commutative analysis is involved. Unfortunately, up to now there does not exist a sufficiently simple exposition of noncommutative analysis which might serve as an introduction to the subject for scientists who are just beginning to get acquainted with this area and that is why this book has been written. It is primarily addressed to those who are not specialists in noncommutative analysis. At the same time, even the experienced mathematician can find in this book many new and interesting topics. Noncommutative analysis gives a new outlook both on quite traditional and modem mathematical topics such as representation theory, operator theory, the theory of (pseudo)differential operators, Yang-Baxter equations and others.