Commutative spaces, as such, have to wait until p. 153 before they make their appearance in this important book on an important subject. Prior to their appearance it is necessary to develop various preliminaries in some detail, including the general theory of topological groups and the representation theory of compact (and locally compact) groups, very substantial subjects in their own right. The wait is well worth it, however, given that the author goes on to develop some very beautiful and substantial mathematics surrounding the subject of commutative spaces, including the generalized Fourier analysis that can be regarded as a raison d’être for the whole subject, particularly Plancherel formulas (and the uncertainty principle). Subsequently the book culminates in a good deal of structure theory. This is scholarship of a high order, presented in a very readable and complete manner.
But what is a commutative space, then, and why is it important? Right off, it is clear that a subject which features a formulation of Heisenberg uncertainty in Fourier- or harmonic analytic terms must owe a great deal to Von Neumann’s seminal work laying out the mathematical foundation of quantum mechanics, in his famous book by that title. Thereafter, historically speaking, quantum mechanics for mathematicians (to coin a phrase?) can possibly be said to have reached a steady state, at least as far as its presentation goes, with Mackey’s book bearing the same title as Von Neumann’s (cf. this reviewer’s on-line review) followed by Varadarajan’s beautifully written Geometry of Quantum Theory. The presence of these players in the game points to some of the major themes of the subject of commutative spaces, to wit, unitary group representations, the Stone-Von Neumann-Mackey Theorem, and sundry methods from Hilbert space theory.
Varadarajan once told me that number theory and physics are but two sides of the same coin; I only came to understand what he meant when I read Weil’s very famous 1964 paper, “Sur certains groupes d’opérateurs unitaires” (Acta. Math. 111). This work deals with what is now called the Segal-Shale-Weil (projective) representation, or just the Weil- or oscillator representation. Despite its specifically number theoretic nature (Weil recasts Hecke’s proof of quadratic reciprocity in representation theoretic terms, for example), it features such objects as Heisenberg groups, Schrödinger representations, and Fock spaces. There can hardly be a more dramatic illustration of the scope of the subject at hand.
With unitary representation theory coming to the fore like this, and with burgeoning connections to the theory of automorphic forms, Weil’s paper was followed in 1969 by Representation Theory and Automorphic Forms by Gel’fand, Graev, and Piatetskii-Shapiro, and, in 1976, by Mackey’s The Theory of Unitary Group Representations , which enjoyed an eventful pre-existence as University of Chicago Lecture Notes. And then there was Harish-Chandra, of course…
Unquestionably, abstract harmonic analysis was on the march, and it progressively expanded its charter to cover spaces of an ever more exotic topological character, to wit, the commutative spaces of the title. This at last brings us to their definition (see pp. 153–154 of Wolf’s book): If G is a locally compact group and K a compact subgroup of G, write KG/K for the usual double coset space and, choosing a normalized Haar measure, define the Banach space L1(KG/K), a natural convolution algebra. If this algebra is commutative, (G, K) is called a Gel’fand pair and G/K is, by definition, a commutative space relative to G. It is in fact the case that if G/K is a Riemannian symmetric space and G qualifies as the largest connected group of isometries of G/K, then G/K is a commutative space relative to G.
Thus, a huge amount of mathematical structure becomes available in this abstract setting, and connections with other parts of mathematics abound. So it is that the second half of Harmonic Analysis on Commutative Spaces covers such topics as commutative Banach algebras, spherical transforms, Pontryagin duality and spectral theorems (for the case of commutative groups), Finsler symmetric spaces, and a substantial analysis of commutative nilmanifolds. Quite a tour de force!
Indeed, Joseph Wolf’s Harmonic Analysis on Commutative Spaces is a splendid source from which to learn this broad and beautiful subject, both for its own sake and with an eye toward application (e.g. to the Langlands program). It is a well-written monograph by an expert in the field and should serve to fill the heretofore un-cast role of a single source for this material. The seminal works by Weil, Mackey, Gel’fand–Graev–Piatetskii-Shapiro, Kirillov, Harish-Chandra, and so on, are still a sine qua non, but a top-quality single volume on the subject of harmonic analysis on commutative spaces is most welcome!
Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.