Question

I am looking for a good book on Topological Groups. I have read Pontryagin myself, and I looked some other in the library but they all seem to go in length into some esoteric topics.

I would love something 250 pages or so long, with good exercises, accessible to a 1st PhD student with background in Algebra, i.e. with an introduction covering necessary background in Functional Analysis.

If possible, I would also like it covering particularly important (in my view) topics:

• emphasize on locally compact groups, but both locally Euclidean and totally disconnected cases;
• Pontryagin duality;
• Kazhdan property T;
• Tannaka reconsruction.

Answer 1

I'm not aware of a book that covers simultaneously Pontryagin duality, property (T) and Tannaka duality. I will refrain from recommending any book on property (T) (guess why?). Apart from Weil's book already mentioned, my favourite ones are:

• for Pontryagin duality: Rudin's "Fourier analysis on groups";

• for functional analytic aspects: Loomis' ``An introduction to abstract harmonic analysis'';

• for representation theory and Tannaka duality (and learning through exercises!): Kirillov's ``Elements of the theory of representations''.

• for group $C^*$-algebras: the second half of Dixmier's ``$C^*$-algebras''.

Answer 2

How about Weil's classic: "L'intégration dans les groupes topologiques et ses applications"? You won't find Kazhdan's Property T nor Tannaka reconstruction, but it treats the other topics deeply and beautifully. Plus, it's good French practice if the 1st-year PhD student needs the practice.

Answer 3

Hewitt & Ross, Abstract Harmonic Analysis vol. 1, 1968

but it seems you didn't want 500 pages

Answer 4

Try An Introduction to Topological Groups by P. J. Higgins (London Mathematical Society Lecture Note Series 15, 1975).

Answer 5

For Tannaka duality of compact groups, you can also have a look at Hochschild's book, "The structure of Lie groups"; it also covers a bit of locally compact group theory if I remember well. For everything except Khazdan, Hewitt & Ross' books are indeed nice (but perhaps a bit too much to digest at once).