What is a good book on topological groups? 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 reconstruction.

 A: 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.
A: Try An Introduction to Topological Groups by P. J. Higgins (London Mathematical Society Lecture Note Series 15, 1975).
A: Hewitt & Ross, Abstract Harmonic Analysis vol. 1, 1968
but it seems you didn't want 500 pages
A: 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 Kazhdan, Hewitt & Ross' books are indeed nice (but perhaps a bit too much to digest at once).
A: I have just read T. Tao's 'Hilbert's Fifth Problem and Related Topics"
and I found it fantastic.
A: 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''.
