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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.
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You might want to take a look at Hewitt-Ross (two volumes). – johndoe Oct 1 '12 at 17:23
should be cw... – YCor Oct 1 '12 at 21:19
250 pages and covering all those topics, while including an "introduction covering necessary background in Functional Analysis"? Really? – Yemon Choi Oct 2 '12 at 0:56
up vote 6 down vote accepted

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''.

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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.

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

but it seems you didn't want 500 pages

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

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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).

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