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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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  • $\begingroup$ 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? $\endgroup$
    – user30035
    Commented Feb 10, 2013 at 23:09
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    $\begingroup$ Does chapter 14 of Wallach's "Real Reductive Groups II" have what you want? $\endgroup$
    – B R
    Commented Feb 10, 2013 at 23:18
  • $\begingroup$ @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 ? $\endgroup$
    – Joël
    Commented Feb 10, 2013 at 23:57
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    $\begingroup$ 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. $\endgroup$
    – Yemon Choi
    Commented Feb 11, 2013 at 0:47
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    $\begingroup$ Dixmier's boom seems to contain everything that I want. Also Wallach's, for what I can judge from google book. Thank you all! $\endgroup$
    – Joël
    Commented Feb 11, 2013 at 2:39

2 Answers 2

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In my humble opinion the best reference is Dixmier $C^*$-algebras. The first half of the book has a very complete explanation of what you need to know about $C^*$-algebras. In chapter 8 he goes over what is the decomposition of a trace for $C^*$-algebras. Then from Chapter 13 on he goes into the theory for a locally compact group. He explains necessary and sufficient conditions for the Plancherel formula to exist (has to be Type I, separable, postliminal, unimodular etc.). He also explains the topology to be given $\widehat{G}$; in fact he gives three different topologies on this set, and shows all of them agree in the case we are interested in—it is just that beautiful of a book. Chapter 18 is the statement of the Plancherel Theorem; the proof essentially is the one in Chapter 8 for $C^*$-algebras. The english version is very good, with very few typos or print mistakes that may confuse you. I have not found a typo or a mistake of any sort in the French version. I think it is a very good book, like reading a novel.

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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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  • $\begingroup$ 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 $\endgroup$
    – Yemon Choi
    Commented Feb 12, 2013 at 1:40

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