On p. 89 of the book under review we find the following:
Our aim is to prove the following results on finite-dimensional representations of compact groups … (1) that every representation is equivalent to a unitary representation, (2) that every representation is completely reducible, (3) the orthogonality relations, and (4) the Peter-Weyl theorem.
With these remarks Chandrasekharan introduces the fourth and final chapter of this compact book on topological groups, indeed on the according representation theory. With the author, now 91, a mainstay at both the Tata Institute and the ETH in Zürich as well as a major figure in number theory, the book enjoys the advantage of a certain terseness, great accessibility, and an implicit focus on the applications of this material to other parts of mathematics.
Clearly Chandrasekharan is keen on getting somewhere fast, and he is willing to dispense with aspects of the theory that don’t serve his purpose. In fact, the book starts off with an “Author’s Note” to the effect that his “aim is a proof of the Peter-Weyl theorem (1927) that every complex-valued continuous function on a compact topological group is a uniform limit of finite linear combinations coming of representation functions coming from irreducible representations.” With that Chandrasekharan appends a list of sources (all classics) and singles out von Neumann and Weil as the originators of the circle of ideas used in the upcoming treatment which, in turn, is based on Ambrose’s 1952 MIT lectures.
And then he is off to the races: the first three chapters deal with, respectively, the obligatory topological preliminaries, a development of the theory of Haar measure on an LCA group, and a discussion of Hilbert spaces and the spectral theorem. Regarding the latter, Chandrasekharan presents a particularly accessible statement in five parts that tells the story very well and leaves little room for confusion: T is a real, completely continuous operator on a Hilbert space with eigenvalues λn; then
- for all positive ε, there are only finitely many λn with |λn| > ε, whence (easy exercise) we get only a countable list of λn, and λn → 0;
- each nonzero eigenvalue has a finite dimensional eigenspace;
- the full Hilbert space is the direct sum of these eigenspaces;
- the closure of T’s image is the sum of the eigenspaces coming from the non-zero eigenvalues; and
- an operator S on the same Hilbert space commutes with T iff S leaves each of T’s eigenspaces invariant.
This explicit enumeration of properties of the eigenvalues of T is obviously very useful and makes for great clarity.
In point of fact, great clarity is one of the virtues particularly in evidence as far as this book is concerned: it is a pleasure to read, and qualifies as a particularly good source for learning this important material — or to use as a text in a course.
Finally, just for good measure, here is the very last sentence of the book (cf. p. 114): “Every locally Euclidean group is isomorphic (group isomorphism and homeomorphism of the space) to a Lie group (Hilbert’s Fifth Problem, 1900). [Proved by D. Montgomery, L. Zippin, and A. Gleason, 1952–1953.] It does not get much prettier than that.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.