What is a good book on topological groups?

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''.


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.


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

but it seems you didn't want 500 pages