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I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they are defined). This topic seems pretty standard to me, but when I needed a basic reference on this (both to check my memories and to be able to cite it in a paper I am writing), I didn't find one.

By the way, the wikipedia webpage "Plancherel measure" should be completely rewritten. There is not even a definition, just a list of examples (and the definition given in the finite case is not compatible with the one given in the compact case). I would be happy to rewrite it, when I have a reference to check the details.

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Joel -- I have a lot of sympathy :-/ One option is just to start reading the complete works of Harish-Chandra and press on from there, but I am really hoping that someone can come up with a better option. At the back of my mind I suspect that you might only be interested in e.g. $G(K)$ for $G/K$ reductive and $K$ $p$-adic or real/complex. Maybe your question would be easier to answer if you just stuck to those cases? Or do you really need something more general? –  user30035 Feb 10 '13 at 23:09
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Does chapter 14 of Wallach's "Real Reductive Groups II" have what you want? –  B R Feb 10 '13 at 23:18
    
@wccanard. Actually the only case I really need is the case of a compact group (a Galois group, so I am on the other side of the Langlands frontier :-) and here of course there are no difficulties. However, I wanted to state a definition in its natural generality, which is whenever there exists a Plancherel measure on $\hat G$. But now I realize I don't know exactly when such a thing is defined. For group of type I ? For all local compact group ? –  Joël Feb 10 '13 at 23:57
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Plancherel for 2nd countable Type I groups can be found, IIRC, in Dixmier's book on C-star algebras - but the description is not explicit. Something like the free group on two generators fails miserably to have a Plancherel theorem, because decomposition as a direct integral of irreducibles is grievously non-unique - this is explained well in Alain Robert's little book on representations of locally compact groups. –  Yemon Choi Feb 11 '13 at 0:47
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Dixmier's boom seems to contain everything that I want. Also Wallach's, for what I can judge from google book. Thank you all! –  Joël Feb 11 '13 at 2:39
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1 Answer

Hartmut Fuehr's book (Abstract harmonic analysis of continuous wavelet transforms, Springer Lecture Notes in Mathematics, Nr. 1863, 2005, X, 193 p., Softcover ISBN: 3-540-24259-7), contains a "-reasonably self-contained- exposition of Plancherel theory", see also http://www.matha.rwth-aachen.de/~fuehr/book.html.

Then there is also Dixmier's book: J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969.

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I had forgotten about Fuehr's book! I am currently having to use some of it as a reference for a paper I am writing, and it is nice to see some of the details (concerning domains of various operators, intersections of various Banach spaces) spelled out rather than just swept under the carpet –  Yemon Choi Feb 12 '13 at 1:40
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