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The Structure of Compact Groups

Karl H. Hofmann and Sidney A. Morris
Walter de Gruyter
Publication Date: 
Number of Pages: 
de Gruyter Studies in Mathematics 25
[Reviewed by
Michael Berg
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This huge book (over 800 pages) is offered as both a primer for the relative novice and a reference book for the expert. This seems like an eminently fair description, although, judging by the authors’ introductory remarks, their original aim was oriented more in the first direction. Indeed, as befits a primer, there is plenty of space allotted to the foundational aspects of the subject. And with a subject like compact groups this is obviously necessary: one doesn’t simply jump in if one hasn’t swum in these waters before.

Hoffmann and Morris stress that their concern is structure theory per se, meaning that even though they take pains to cover the allied themes of representation theory and harmonic analysis at the proper foundational level, and in fact slightly beyond, they forego any more intensive or extensive treatment of these themes, being keen to get on with structure theory. Of course this is not a problem, given that representation theory of all flavors, as well as harmonic analysis, is abundantly covered in any number of other texts, many very well established. The standard works by Kirillov and Gel’fand-Graev-Piatetskii-Shapiro come to mind immediately, for example.

Additionally, The Structure of Compact Groups treats Lie theory only as a (major) tool in the service of the authors’ classification theoretic goals, but here the same observation as above applies even more emphatically: Lie theory is surely more than abundantly represented in the literature. Nonetheless, Hoffmann and Morris do suggest that parts of the book under review can be used for “self-contained courses on linear Lie groups, on compact Lie groups, and on locally compact abelian groups.” This is certainly pedagogically sound.

Furthermore, The Structure of Compact Groups is distinguished by the unusual and desirable feature of purposely circumventing the usually pervasive tactic of approximating compact groups by Lie groups, to the point that projective limits are explicitly eschewed (despite pro forma coverage of this material in the book’s section on homological algebra) in order to be able to get at classification problems via methods that are “free… of all dimensional restrictions, in particular, of the manifold aspects of Lie groups.” This is obviously of great interest to the insiders.

The Structure of Compact Groups sports twelve chapters, from the relative basics (viz. Chapter 3, “The ideas of Peter and Weyl,” and Chapter 4, “Characters”) to hairy-chested stuff like Chapter 12, “Cardinal invariants of compact groups.” There are also four useful appendices, on abelian groups, covering spaces, category theory, and topological groups, all adding up to a titanic work, now in its second evolutionary stage (the first edition appeared in 1998: the book under review is the “2nd Revised and Augmented Edition).

Finally, the book is not without humor: consider, for example, the footnote on p. 472, dealing with a walrus. We leave this to the interested reader.

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

The table of contents is not available.