C\({}^*\)-algebras and operator theory. (English) Zbl 0714.46041

Boston, MA etc.: Academic Press, Inc. x, 286 p. $ 44.50 (1990).
Because of its rapid development in recent years the theory of operator algebras has become more and more inaccessible and diverse. Now, here is an excellent modern introduction to the basic theory of \(C^*\)-algebras. It is very well suited for an introductory course on the subject. In fact, at the moment, I am using the book for a course and I am really pleased with the exposition and the way, the arguments are set up. The choice of material is very adequate. Most of the important and basic notions are treated in an elegant and clever way. The exercises complement the text nicely.
Among the topics covered are: Banach algebras, Gelfand representation. Compact operators, Fredholm operators, Toeplitz operators, spectral theorem. Representations and positive functionals for \(C^*\)-algebras, multipliers, ideals and hereditary subalgebras. Von Neumann algebras, double commutant theorem, Kaplansky density theorem. Inductive limits, AF-algebras. Tensor products, nuclear \(C^*\)-algebras. K-theory, long exact sequence, Bott periodicity.
None of the topics is treated really in depth. However a student who has worked through this book, has a very good basis for starting research on \(C^*\)-algebras. At the same time the book presents an introduction to the algebraic theory of operators in Hilbert space.
A few things, that one would like to see included (may be in the next edition) to round up the book completely, are missing: crossed products and group \(C^*\)-algebras, classification into different types of von Neumann algebras, existence of a trace on von Neumann algebras of type II, some non-trivial computations of K-groups.
Reviewer: J.Cuntz


46Lxx Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46H05 General theory of topological algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)