This book provides exposition of the basic theory of quantum bounded symmetric domains. The area became active in the late 1990s at a junction of noncommutative complex analysis and extensively developing theory of quantum groups. It is well known that the classical bounded symmetric domains involve a large number of nice constructions and results of the theory of $C^*$-algebras, theory of functions and functional analysis, representation theory of real reductive Lie groups, harmonic analysis, and special functions. In a surprising advance of the theory of quantum bounded symmetric domains, it turned out that many classical problems admit elegant quantum analogs. Some of those are expounded in the book. Anyone with an interest in the subject will welcome this unique treatment of quantum groups.

The book is written by a leading expert in a very clear, careful, and stimulating way. I strongly recommend it to graduate students and research mathematicians interested in noncommutative geometry, quantum groups, $C^*$-algebras, or operator theory.

—Vladimir Drinfeld, University of Chicago

Readership

Graduate students and research mathematicians interested in representation theory and noncommutative geometry.