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Publisher:

European Mathematical Society/American Mathematical Society

Publication Date:

2006

Number of Pages:

302

Format:

Hardcover

Series:

EMS Textbooks in Mathematics

Price:

58.00

ISBN:

3-03719-016-7

Category:

Textbook

[Reviewed by , on ]

Kenneth A. Ross

08/31/2006

As the title suggests, this book contains the basic structure theory of locally compact groups. There is considerable overlap with [HR1], but in many ways this book reflects a more modern outlook. For example, the author smoothly uses exact sequences and commutative diagrams wherever appropriate. Also, the reader is encouraged to think in terms of various categories of Locally Compact (LC) groups, and Chapter E is devoted to this topic.

The last two short chapters are good introductions of topics that readers may wish to pursue. Chapter G concerns topological semigroups, a special interest of the author, while Chapter H gives a survey of Hilbert's Fifth Problem.

Unlike [HR1], this book is primarily written for students. The modern outlook and the friendly style and format are compatible with this goal, and I am confident that most students would benefit greatly from studying this book. I have at least two concerns for such an audience, however. The text tends to dwell on technical issues and strive for maximal generality, for example, by including non-Hausdorff groups. Also, the examples illustrating new concepts are often trivial.

In the preliminaries on general topology, I applaud the author's use of both filters and nets, as well as the avoidance of subnets. Stroppel proves many of the basic topological results, but avoids the proof of Tychonoff's theorem on product spaces. A nice rather intuitive proof by Paul Chernoff (1992), and improved by Charles Pugh (2003), uses nets and accumulation points but not subnets; see http://darkwing.uoregon.edu/~ross1/SetTheoryAppendix.pdf.

In section 3, left and right translations in groups are first illustrated in the setting of the additive group of real numbers! The group axioms in 3.1, and the definition of inner automorphisms in 3.10, are well disguised by awkward notation. Section 13 studies "module functions"; as far as I know, all other authors call them modular functions. A good exercise would be: Module functions on totally disconnected LC groups take only rational values. An example of a non-unimodular totally disconnected LC group also would be nice.

The proof of 9.13 is very slick, though a reference to 1.5(d) would have been helpful. This allows the author to prove 9.15 much more efficiently than is done in [HR1 , 26.5].

The author proves a nice result in 23.13 which is new to this reader: Closed subgroups of compactly generated locally compact Abelian groups are compactly generated. In section 21, he notes in passing that closed subgroups of general compactly generated LC groups need not be compactly generated. This is also new to this reader; an example or reference would have been helpful.

In connection with the Pontryagin duality theorem in section 22, there is also a probabilistic approach due to K. R. Parthasarathy, *Periodica Mathematicae Hungarica* **2** (1972), 21-26.

Other comments. The use of lambda for *right* Haar measure is unfortunate, especially since the adjective "right" is often omitted. In 23.24, parts (b) and (c), the groups *A* need to be in **LCA** .

The book includes two helpful indexes, but here are some missing entries: For the Index of Symbols: *D*_{ε} 20.1; *G*_{1} connected component, 4.9; *i _{g}* inner automorphism, 3.10; ω

**Reference:**

[HR1] Edwin Hewitt and Kenneth A. Ross, *Abstract Harmonic Analysis I*, Springer-Verlag 1963, second edition 1979.

Kenneth A. Ross (ross@math.uoregon.edu) taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

- Preliminaries
- Topological groups
- Topological transformation groups
- The Haar integral
- Categories of topological groups
- Locally compact Abelian groups
- Locally compact semigroups
- Hilbert's fifth problem
- Bibliography
- Index of symbols
- Subject index

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