What a fabulous title! How does one bluff in the game of spectral mapping theorems? Well, what game is it, i.e. what’s the bigger picture? The answer is that these theorems are central to the study of commutative Banach algebras, and the author’s stated goal is to “describe the spectral mapping theorem in one, ‘several,’ and ‘many’ variables.” And what do these things mean? Well, a first pass at the spectral mapping theorem comes from linear algebra: say we’re working with finite dimensional vector spaces and in this context a given matrix A has eigenvalues \(\lambda_1,\lambda_2,\dots,\lambda_n\); then, for any polynomial \(P\), the matrix \(P(A)\) has eigenvalues \(P(\lambda_1),P(\lambda_2),\dots,P(\lambda_n)\) (see, e.g. the ubiquitous Wiki-source on this business). Thus the book under review goes at a generalization of this result, with, of course, the word “spectrum” now taking on a more sophisticated meaning than “set of a matrix’s eigenvalues,” to the setting of Banach algebras. Not surprisingly the name Gel’fand figures on many pages in the book.

Indeed, right from the start, Harte presents the object of his study in the following evocative way.

Suppose \(ab=ba\in A\), a commuting pair of Banach algebra elements … ; then \(\sigma(a+b)\subseteq \sigma(a) + \sigma(b); \sigma(ab) \subseteq \sigma(a)\sigma(b)\) … One way to prove this is via Gel’fand’s Theorem [involving maximal ideal spaces, of course] … [But a m]uch sweeter [result] would be a joint spectrum argument enabling us to write \(\sigma(a+b) = \{\lambda+\mu: (\lambda,\mu)\in \sigma(a,b)\}; \sigma(ab) = \{ \lambda\mu : (\lambda,\mu)\in \sigma(a,b)\}\) with \(\sigma(a,b)\subseteq \sigma(a)\times\sigma(b)\). That, in a nutshell, is what these notes are all about.

As always, \(\sigma\) stands for spectrum, the notion of “eigenvalue set” on steroids.

This book is, by design and charter, very terse (not much over 100 pages: it’s truly a Springer Brief), and is therefore vouchsafed for the more seasoned reader: Harte says that “[a]s background we need to introduce the basic algebraic systems, including semigroups, rings, and linear algebras,” but I think it’s the wise reader who comes to this game with more preparation and experience. To be sure, the first three chapters of the book are devoted to “algebra,” “topology,” and “topological algebra,” but it’s already around p. 40 that we get, in quick succession, “spectral topology,” “normed algebra,” and “Banach algebra.” It’s a very rapid ascent.

That said, it’s a very nice book indeed, given that the reader who survives the g-forces encountered in the opening chapters gets to see some beautiful things in the last three chapters, including a load of spectral theory properly so-called (always worth the price of admission) and generalizations of the material (all geared toward spectral mapping, of course) to “several” and “many” variables. Here, by Harte’s definition, “several” refers to the setting of several-variable polynomials, while “many” means that we’re indeed going over to infinitely many variables: “the passage from finite *n-*tuples … of linear algebra elements to more general systems [indexed on a possibly infinite index set].”

It’s all a lot of mathematical fun, on the sophisticated side, to be sure, but presented well and in a very good, if compact, style.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.