# Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstract 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.

## 2 Answers

This is an incomplete answer, and the OP seems to have left MathOverflow several years ago, but since the only answer that has been posted thus far gives an incorrect "answer" to the first question, which has not been stricken out in anyway, I would like to post a reference which does answer the first question of the OP --- and perhaps reviving this question will prompt other users who know the literature better than me to point out other sources!

Regarding Q1: contrary to what the earlier answer says (at time of writing), 18.8.1 in Dixmier's book does not address cases where $$G$$ is not second-countable. (There seems to have been confusion arising from the usage, standard at the time, of "separable" to mean "second countable", as discussed in comments to that earlier answer.) However, in the notes to Chapter 18 we find (English translation):

18.9.2. The Plancherel formula can be generalised to non-separable postliminal unimodular locally compact groups. [452]

where [452] is the paper

J. Dixmier, Traces sur les $${\rm C}^\ast$$-algèbres. Ann. Inst. Fourier 13 (1963) 219–262. NUMDAM link

The precise statement can be found as Théorème 3 (Section 16).

As already mentioned, "non-separable" here means "not 2nd countable"; postliminal (postliminaire) means GCR in the sense of Kaplansky. Note that for $${\rm C}^*$$-algebras, the equivalence of GCR with Type I (Glimm's theorem) relies on a separability assumption for at least one direction.

A quick look at Dixmier's paper shows that his method is to consider the Hilbert algebra $$L^1(G)\cap L^2(G)$$ and the induced (Plancherel) weight on the von Neumann algebra generated by this Hilbert algebra; this weight is tracial and its restriction to the group $${\rm C}^*$$-algebra $$A$$ is a densely-defined lower-semicontinuous trace on $$A$$; he then appeals to earlier results in his paper concerning such traces, where the GCR assumption on $$A$$ is used to bypass the disintegration-of-von-Neumann-algebras approach of Segal and others (which did require $$L^2(G)$$ to be separable).

Regarding Q2: The reliance on traces may explain, to some extent, why this approach cannot help with non-unimodular groups. I admit I've not looked up how one bypasses non-unimodularity for 2nd-countable Type I groups; it might be relevant that $${\rm C}^*$$-algebras which do not support any densely-defined faithful tracial state can have WOT closures that support faithful normal semifinite traces in the sense of von Neumann algebras. The real $$ax+b$$ group is an example of such a phenomenon, and the Plancherel theorem for that group can actually be found in the Folland book which the OP mentions.

Regarding Q3: I suspect Kirillov is alluding to decompositions of the left regular representation of $$G$$ as a direct integral of factor representations, which can be seen as a weakened subtitute for a decomposition as a direct integral of irreducible representations. I think some Lie group cases that are not Type I were worked out by various authors in the 1970s/1980s but I am not familiar with the literature. Folland's book (the one mentioned in the original question) has some commentary on this topic but no proofs.

Coda: I am a bit surprised to see that Dixmier's paper has relatively few citations, even though it seems to have been written at the same time as several other "foundational" papers of his. Perhaps this reflects the fact that only the second-countable case (a.k.a. separable Type I $${\rm C}^*$$-algebras) made it into his book, and it is the book which has become a standard reference rather than the series of papers that went into his book.

To be honest: the lack of citations, even in papers which make explicit reference to Plancherel-type theorems for non-2nd-countable groups that are "close to abelian or compact", does make me slightly concerned that there may be problems with Théorème 3 in Dixmier's paper, perhaps known to experts but not recorded in the literature. Of course I hope that this is not the case!

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.

• 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
• My point was that Dixmier requires second-countability/separability (I do not know whether they are equivalent for locally compact groups—I’ve never seen such a statement) for the Plancherel formula, too. Thus it is not a reference telling us that this condition can be dropped (which is claimed in the book by Kirillov). Regarding “What is the suggested analogon you have in mind?” In which situation? – The User May 17 '13 at 10:44
• 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
• 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:( – Marc Palm May 17 '13 at 14:49
• @TheUser Bourbaki's definition of "separable" was what we now call "second countable" (General Topology IX.2.8 Definition 4), although restricted to the case of metrizable spaces, and he proves that it is equivalent to "having a countable dense subset" in Proposition 12 of the same section, which is given no name (admittedly, separability is not that useful outside the metrizable case). So French authors of that period, such as Dixmier, will generally follow that terminology, as well as others ("compact" to mean "compact Hausdorff" etc.). – Robert Furber Aug 23 '19 at 21:28