Book review

This book is dedicated to the construction of mathematical models for a number of physical systems in the language of families of commuting self-adjoint operators. The spectral theory of finitely many commuting self-adjoint operators is treated in the book of M.Sh. Birman and M.Z. Solomyak, The spectral theory of self-adjoint operators in Hilbert spaces, Leningrad University Press, Leningrad-1980 (in Russian), and the applications of self-adjoint operators in modern mathematical physics is presented largely in the four volumes of the book of M. Reed and B. Simon, Methods of modern mathematical physics, Academic Press, from 1972 to 1978. The spectral theory for arbitrary systems of commuting self-adjoint operators was treated by Yu. A. Berezanskii, Self-adjoint operators in spaces of functions of infinitely many variables, Naukova Dumka, Kiev 1978, English translation, Amer. Math. Soc., Providence R.I., 1986. The author concentrates in this book on a special case of such systems, namely countable collections of commuting self-adjoint operators and their joint spectral properties. Such collections have a lot of additional properties which yield a simplification of the formulations of general theorems as well as a number of new assertions, similar to their finite dimensional analogues. There exists a comparable situation in the study of random sequence, because one can consider a commuting family of self-adjoint operators as a "non-realized" random field. The correspondence between problems in the theory of random processes and the theory of infinite families of commuting operators is treated in the fourth part of the book.

The theory of countable families of commuting self-adjoint operators is also connected with the theory of unitary representations of inductive limits of commutative locally compact simply connected Lie groups, a topic treated in the second chapter of the book. These theories are then applied in Chapter 3 to the study of differential operators with constant coefficients on functions of countably many variables.

The book is divided into four parts and twelve chapters: Part I, Families of commuting normal operators; Part II. Inductive limits of finite-dimensional Lie algebras and their representations, Part III. Collections of unbounded self-adjoint operators satisfying general relations; and Part IV. Representations of operator algebras and non-commutative random sequences.

The theory of *-algebras and their representations begins only in the fourth part, where non-commutative probability theory, non-commutative random sequences and non-commutative dynamical systems are introduced, studied and applied to problems in quantum physics.

The result is a fine book, accessible to people familiar with a basic university course in mathematics, including an exposition of the theory of unbounded self-adjoint operators. Some more special results needed for the reading of the book are measure theory on sequence spaces, unitary representations of Lie groups and Lie algebras and some fundamental facts of the theory of *-algebras and their representations.

The book appears in excellent typographical conditions, contains a large bibliography and certainly will become and indispensable tool for all interested in mathematical models of quantum systems.