Non-Commutative Analysis

What an unusual book this is! It purports to be a Functional Analysis text but it is unlike most this reader has ever read. Rather than the typical “theorem, proof, application” sequence this treatment starts with the applications and then unravels the mathematical approach (along the way including historical remarks and further motivation). No student (metaphorical or actual) will ever raise the plaintive cry “But professor, when will we ever use this?”

The title of this unusual book is meant to indicate mathematical subjects that share territory with the classic non-commutative paradigm at the heart of quantum theory: the theory of operators in Hilbert space, harmonic analysis, numerical analysis, partial differential equations, representation theory, ergodic theory, Lie groups and stochastic processes and probability. All of these, it should be noted are given meaningful introductions so that the enquiring undergraduate can find a starting point into this conglomeration of fields. That may sound unbelievable but the authors accomplish this quite nicely.

Most traditional Functional Analysis texts typically begin with coverage of the “big four” results: the Hahn-Banach Theorem, the Open Mapping Theorem, the Uniform Boundedness Principle and the Closed Range Theorem. This book only mentions Hahn-Banach (with a proof in an appendix). The majority of emphasis here is on the spectral theory of unbounded operators on a Hilbert space. This is due to the modern view of quantum physics that lies behind nearly everything in this book (momentum, for instance, is really the derivative operator in disguise and thus unbounded even on simple bounded sets consisting of trigonometric functions). This is a rather large departure from typical FA classes which often begin with the very abstract (e.g. topological vector spaces) and only include applications as an afterthought.

Several chapters spell out the background ideas as “elementary facts” (including the whole concept of duality).The movement from idea to idea is fast and hence the questions often involve important connections. For instance, one of the first questions (2.1, page 31) asks students to show that the sequence of Cesàro means of a contraction operator (norm less than or equal to one) converge to an operator (undefined) \(P\) which satisfies \(P^2=P\). Given that there is no real talk of convergence for operators (and no examples) before this (other than a definition of weak*-convergence for operators) I was left wondering what a student would make of it. The next few pages introduce Schwartz’s Distribution Theory, Gelfand Triples and the Axiom of Choice! This is clearly not a book for the faint of heart.

If I were a physics major with minimal exposure to Real Analysis I would undoubtedly need a lot of support from the teacher and perhaps a few enlightening examples to work through. This is where the book feels incomplete to me — some collected answers or hints in the back of the book would go a long way towards helping one use this for independent study. To be fair, the book evolved from a graduate level elective (which allows some senior undergraduates!) designed to serve students pursuing both mathematics and physics, so much of this support is probably supplied in class or recitations. I still find it rather incredible to believe even a beginning graduate student in mathematics would be able to read a paragraph description of the dual to a Banach space and then knock off a proof of its completeness as an exercise.

The latter chapters on Lie Groups and Representations seem more designed for a new reader and are better paced. The last few chapters go introduce up-to-date topics that I have not seen represented in any mathematical physics texts: unbounded graph Laplacians and Reproducing Kernel Hilbert Spaces showcase the advances that mathematics continues to make in regard to the great enterprise of developing models for physics.

The book gets so much absolutely right (appendices that contain guides to the important names behind the math, reviews of other FA books, pithy quotes at the start and ends of chapters by giants of the field — this book is obviously a labor of love) that is a pleasure to “surf” it. The nature of modern mathematical physics, of course, is that it involves the use of many far-flung fields of algebra and analysis. This book could easily be used as a blueprint for an entire mathematical physics major in this regard. To read it is to appreciate with pleasure the grand scope of that enterprise.

Jeff Ibbotson is the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.