This is one of those books whose Preface is truly a perfect (and tantalizing) preview of what lies ahead, and it has the virtue of being terse and to the point, much like the book it introduces. Thus, Arveson, who has “given [the corresponding] lectures several times in a fifteen-week course … normally taken by first- or second-year graduate students with a foundation in measure theory and elementary functional analysis,” presents a tightly structured whole, fitted into an orbit of around 130 pages, and provides the reader with “many deep and important ideas [that] emerge in natural ways.” He is also, from the start, correspondingly explicit in characterizing his objective as “[presenting] the basic tools of modern analysis within the context of what might be called the fundamental problem of operator theory: to calculate spectra of specific operators on infinite-dimensional spaces, especially operators on Hilbert spaces.” So we are on familiar ground, and see around us a beautiful landscape indeed. There is no doubt that the spectral theorem is one of the deepest and most elegant results in all of mathematics, and its importance is hard to overstate. Arveson notes its connection to “the mathematical foundations of quantum physics, non-commutative K-theory, and the classification of simple C*-algebras [as] three areas of current research activity that require mastery of the material presented here.”
Fair enough. What, then, is this material (which, of course, we all have an inkling about already, since functional analysis is part of our common heritage)? Well, Arveson, splits it into four chapters, respectively, “Spectral theory and Banach algebras,” “Operators on Hilbert space,” “Asymptotics: compact perturbations and Fredholm theory,” and finally, and suggestively, “Methods and applications.” Certainly no one would disagree with his choices for the first two chapters (and I particularly like his singling out of the regular representation of a Banach algebra already on p.11), and his third chapter is on target, too: a Fredholm operator is, by definition (cf. p.92), “[a] bounded operator … on a Banach space [whose kernel] is finite dimensional and [whose image] is a closed subspace of finite codimension in [the target space]” — a ubiquitous player in not just pure functional analysis, but a lot of other areas of mathematics. Arveson’s characterization is entirely fair.
In the fourth chapter, “methods and applications,” we start off with maximal abelian von Neumann algebras, go on to Toeplitz theory, hit the index theorem for continuous symbols (presently including Fredholm operators in a Toeplitz C*-algebra), and finish up with the G(elf’and-)N(aimark-)S(egal) construction and the Gel’fand-Naimark theorem vis à vis the existence of states in a unital C*-algebra. And, yes, the definition of “state” is connected to quantum mechanics: “a positive linear functional [taking the unit to 1]” (p.122). The index theorem just mentioned is indeed of the same flavor as the Atiyah-Singer index theorem; if the latter connects a topological invariant to a geometric one, the former (Arveson’s choice) connects the index of a Fredholm operator (i.e. the difference between the dimensions of its kernel and cokernel) to a winding number defined on the complex-valued continuous functions on the unit circle possessing no zeroes: analytic data vs. (ostensibly) topological data. This illustrates (or reminds us) of the gorgeous fact that spectral theory is amenable to being developed as an analogy of sorts with the method of power series and Cauchy theory in complex analysis — expand (I-λT)-1, T being an operator on a Hilbert space and λ a singular value, as a power series in T and you’re off to the races. Additionally it leads the reader to index theory in a relatively painless fashion and this fits with Arveson’s stated goal to prepare his pupils for “areas of current research activity that require mastery of the material presented here.”
Little more needs to be said about this excellent book: it has plenty of good exercises, it is well written, and reaps the benefit of coming from the author’s experience with this important material in his graduate courses at Berkeley. It is indeed a very good textbook in a fundamental and centrally important subject.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.