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The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a square-integrable function on the group from its Fourier transform by taking traces and integrating with respect to the Plancherel measure (alternatively: we can decompose the regular representation into irreducible representations using the Plancherel measure).

In Representation Theory and Noncommutative Harmonic Analysis by Alexander Kirillov the same theorem is stated but without the restriction “second-countable” (theorem 6.12). It is just a survey book—there is neither a proof nor an explicit reference. First question: Do you know a reference for this theorem?

Regarding non-unimodular groups: Duflo and Moore proved that the Plancherel formula still works for non-unimodular groups, but you have to introduce some additional unbounded, positive, “semiinvariant” operators to scale the stuff correctly (see this paper). They require the group to be of type 1 (of course) and second countable.

Second question: Is it known whether this works for non-second-countable groups? Is there any point where second-countability is thought to be crucial?

Third question: Kirillov also mentions generalisations to non-type-1 groups. Then it is not enough to consider irreducible representations, but according to him there is a similar statement. Do you know what theorem he means and do you know any reference? These are his words:

Another generalization is possible for groups which are not of type I. In this case, the integral on the right hand side of the formula is computed over the larger space $\tilde{G}$ and the ordinary trace is replaced by the trace in the sense of the corresponding factor.

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1 Answer

Answer to the first question: Jacques Dixmier, Les C-algèbres et leurs représentations. Section 18.8.1

Comment on the second question: I actually believe a decomposition of von-Neumann algebra into factors is only available for seperable vNas, which should be for the right regular representation equivalent to the group being second countable.

Comment on the third question: I have no idea what could be meant. But the decomposition into factors will not be unique (this is probably what you mean with not enough to consider irreducible representations) and I don't even know what kind of traces should be involved. So for me, it seems unreasonable to expect something useful in this context, which has similar applications as the Plancherel formula.

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Thank you. Dixmier requires “separability” (does he mean second-countability? I have seen people using these words synonymously), too. Why don’t you think that your argument regarding the second question can be applied to the first one, too? –  The User May 17 '13 at 8:53
    
Regarding the third question: I have added a quote by Kirillov. And I have found this paper (ams.org/journals/tran/1962-104-02/S0002-9947-1962-0139959-X/…) regarding decompositions with respect to this $\tilde{G}$. However, in the formula (theorem 2) there do not appear any traces and there is no relation to any kind of Fourier transform given—the proposed decomposition is very generic. The paper seems to be too old to be a proper generalisation, too. –  The User May 17 '13 at 8:57
    
No, I am saying the Hilbert space $L^2(G)$ is seperable in the sense that they have a countable orthonormal basis iff $G$ is second countable. Now, how do you define a state on say $C_c^\infty(G)$ or $C_c(G)$ from a representation $\pi$ if $\pi(\phi)$ is not Hilbert Schmidt, trace class or something analogous. What is the suggested analogon you have in mind? Sure, the integral decomposition exists, but is not unique and the unitary dual is not a nice space anymore. For type 1 e.g. it will be almost Hausdorff. –  plusepsilon.de May 17 '13 at 9:47
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For your information (I have looked it up): Separability and second-countability are not equivalent for locally compact groups: The compact group $\mathbb{T}^{\omega_1}$ is not first-countable, but separable. I guess that Dixmier’s definition of the word “separable” is “second-countable” (I have seen that before in harmonic analysis)—thus it is probably the same condition as Folland’s. –  The User May 17 '13 at 14:28
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Ah okay, first countability is necessary and sufficient for having a metric in a locally compact group. So correction: first countable implies second countable for lc groups if seperable:( –  plusepsilon.de May 17 '13 at 14:49
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