The author of a well-known monograph “An invitation to \(C^*\)-algebras” on \(C^*\)-algebras in the Springer Graduate Text Series (1976; Zbl 0344.46123) now proposes a book on spectral theory in the same series for the same readership. Accordingly, he considers resolvents and spectra right from the beginning in the abstract setting of complex Banach algebras, but several sections (Subsections 1.2 and 2.8, the entire Chapter 3, and some parts of Chapter 4) are of course concerned with the classical example of bounded linear operators on complex Hilbert spaces.
The book consists of 4 chapters: 1. Spectral Theory and Banach Algebras (origins of spectral theory, the spectrum of an operator, examples of Banach algebras, the regular representation, the general linear group, spectra of elements of a Banach algebra, spectral radius, ideals and quotients, commutative Banach algebras, \(C(X)\) and the Wiener algebra, spectral permanence theorem, analytic functional calculus); 2. Operators on Hilbert Spaces (operators and their \(C^*\)-algebras, commutative \(C^*\)-algebras, continuous functions of normal operators, spectral theorem and diagonalization, representation of Banach \(C^*\)-algebras, Borel functions of normal operators, spectral measures, compact operators, adjoining a unit to a \(C^*\)-algebra, quotients of \(C^*\)-algebras); 3. Asymptotics: Compact Perturbations and Fredholm Theory (the Calkin algebra, Riesz theory of compact operators, the Fredholm index); 4. Methods and Applications (maximal Abelian von Neumann algebras, Toeplitz matrices and Toeplitz opeators, the Toeplitz \(C^*\)-algebra, index theory for continuous symbols, some \(H^2\) function theory, spectra of Toeplitz operators with continuous symbols, states and GNS constructions, existence of states: the Gel’fand-Naimark theorem).
The book is well-written and provides a large variety of results, ranging from the historical roots to the frontiers of contemporary reaseach. Of course, the contents of the last chapter is very special and reflects the author’s personal taste, the reviewer would have preferred some more application-oriented results (and much more examples!) on the use of spectra in classical quantum mechanics. So the title may be somewhat misleading for students who expect a basic course. However, the book may be of interest for those who have already got in touch with classical spectral theory during a course on functional analysis and operator theory, and want to learn something about the interconnections of spectra with abstract fields like \(C^*\)-algebras or modern \(K\)-theory.