Graduate Studies in Mathematics. 15. Providence, RI: American Mathematical Society (AMS). xv, 398 p. $ 59.00 (1997).

This is the second edition of the well-known advanced textbook (see

Zbl 0518.46046,

Zbl 0831.46060). The book consists of 5 chapters (“Linear spaces”, “Basics of Hilbert spaces and linear operators”, “Banach algebras”, “Elementary $C^*$-algebra theory”, “Elementary von Neumann algebra theory”), Bibliography, Index of notations, and Index.
In the fifties L. V. Kantorović wrote that functional analysis can be divided into five directions: “Analysis in Hilbert spaces”, “Analysis in Banach spaces”, “Analysis in topological linear spaces”, “Analysis in ordered linear spaces”, and “Theory of Banach algebras and representations”. The treatise by R. V. Kadison and J. R. Ringrose is one of the best among books devoted to the first half of the fifth direction. In the first volume one can find basics of Banach and Hilbert spaces and the theory of linear operators between them, tensor products and Hilbert-Schmidt operators, elements of general theory of Banach algebras including spectral theory and holomorphic functional calculus, theory of positive linear functionals in $C^*$-algebras, projection technique and construction in von Neumann algebras. Although the main aim of this book is to give elements of non-commutative real analysis, the theory of commutative (abelian) algebras is presented with exhaustive completeness. In general, the description of the mathematical theory presented in this book is clear and accurate; exercises given in each chapter make the essential supplement to the basic text. The book can serve as a textbook for a one semester course of elementary functional analysis (Ch. 1-3), of $C^*$-Banach algebras (Ch. 3-5), or a two-semester course of Banach algebras; however, the book can also be used for individual study as well as courses or seminars. Undoubtedly, the acquaintance with this book is useful for all teachers in functional analysis and researchers in the corresponding field.