Long ago, when I was just starting out as a graduate student in number theory, I had occasion to take Leo Sario’s course in commutative Banach algebras. This highly specialized course was in need of “warm bodies,” its famous instructor notwithstanding, and I was accordingly recruited by a friend who was one of Sario’s PhD students. At that point in my student career I had been a fellow traveler, if not a true believer, in such areas as mathematical logic and hard analysis, and functional analysis was even more peripheral to me, I fear. I came to Sario’s commutative Banach algebras in a nigh-on tabula rasa condition, hopeful that there would be much more commutative algebra than Banach, so to speak.
Of course, I was wrong, but happily so (in retrospect): in his marvelously old-world style Sario presented a beautiful course that has left its traces on my memory for going on thirty years. It was a wonderful experience to encounter spectra confined to the unit interval, to meet the Shilov boundary, and to hear the names of Gelfand and Shilov everywhere and all the time. Soon Professor Sario became one of my favorite professors and I fondly recall his cultured erudition (a Finn to the core: he taught me the correct pronunciation of “Nevanlinna,” for example) as well as a sense of functional analysis that would serve me well down the road, even in connection with some aspects of my own doctoral work on modular forms.
Accordingly it is a pleasure to return to commutative Banach algebras in the context of Eberhard Kaniuth’s book, A Course in Commutative Banach Algebras, the book under review. Its lay-out already conveys the author’s good (and effective) conception of how his material should be presented: he packs the meat of the subject, Gelfand theory, the Shilov boundary, regularity, and spectral synthesis and ideal theory, in between an opening chapter titled, “General Theory of Banach Algebras,” and a sextette of very apposite appendices (grouped into a single “Appendix”) including coverage of functional analysis generalities and Pontryagin duality. This testifies to the fact that the reader, or the student, should already be conversant with more basic functional analysis, and in this connection, à propos, Barbara MacCluer’s recent book, Elementary Functional Analysis, reviewed in this column not long ago, should serve well indeed. Thus, Kaniuth’s book is properly suited not to a rookie graduate student but to a rather more mathematically mature exhibit.
On the other hand, there is something about commutative Banach algebras, or even functional analysis in general, that is so appealing and inviting that rushing into the subject, and the book under review, is almost irresistible. And, to be sure, the material in Kaniuth’s text is well suited to such virtuous impetuousness. The definitions are cogent and clear, the theorems are stated well and come equipped with clear and well-written proofs, and the mature and energetic reader can always consult the appendices, when gaps in preparation rear their ugly heads.
It is proper, however, to warn the potential reader at this stage that Kaniuth injects a good deal of relatively sophisticated material into his presentations of even the main themes of the subject: a superficial single reading of the book will not do it justice. And, to boot, A Course in Commutative Banach Algebras is populated by a host of examples and exercises that cover the spectrum (if I may be pardoned an obvious pun) from routine to fruity, and are not to be missed.
Finally, Kaniuth goes the extra mile by prefacing his chapters with cogent introductions presenting both a good overview of what is to follow and a sketch of the larger context; additionally each chapter is capped off by a closing section titled “Notes and References,” containing huge amounts of all too tantalizing information (Kaniuth plays the game very well indeed). What else is there to say? A Course in Commutative Banach Algebras is a wonderfully crafted book full of fine scholarship, dealing with an intrinsically beautiful and important subject. Ever since Sario I’ve liked this subject a great deal; Kaniuth’s book dramatically intensifies the emotion.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.