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It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof of this claim in the literature (including the papers defining LCQGs). Could anyone point me to an appropriate reference?

In particular, my problem is as follows. The C*-algebra of a LCQG $\mathbb{G}$ encodes the topology of the underlying group in its spectrum $G:=Sp(\mathbb{G})$. I'd like to use the Riesz-Markov representation theorem to get a Borel measure on $G$. For that I need to restrict the Haar weight to $C_c(G)$. What ensures that the restriction is finite? My first idea is to use compact operators of $C_0(G)$ acting on the GNS representation...

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There are many ways to prove this. My personal favorite is the following argument. To every locally compact quantum group is associated a multiplicative unitary in the sense of Baaj and Skandalis. When the underlying C*-algebra or von Neumann algebra is commutative, then the multiplicative unitary is commutative (still in the sense of Baaj and Skandalis). In Theorem 2.2 of https://doi.org/10.24033/asens.1677 Baaj and Skandalis proved that every commutative multiplicative unitary comes from a locally compact group. In https://doi.org/10.1016/S1631-073X(03)00034-7 they gave a more easy proof of the same result.

Also note that if you use a von Neumann algebra definition for locally compact quantum groups, then the proof will never be fully trivial, because the result then contains the classical Weil theorem saying that a "measured group" with an invariant measure must be locally compact.

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  • $\begingroup$ Thank you, that was exactly what I was looking for and helped a lot! As a small side question, do you maybe know if there is any theory of 'locally compact measure spaces' without the group structure, so without the Weil theorem in the classical limit? (If I were to guess something involving a von Neumann algebra $L^\infty(\mathbb{X})$ with a weight + some compatible unique C*-algebraic structure) $\endgroup$
    – szantag
    Commented Jun 3 at 21:27
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    $\begingroup$ @szantag I may misunderstand the question, but I would say that the theory of weights on C*-algebras, and in particular KMS weights on C*-algebras, could be considered as a noncommutative theory of locally compact spaces with a measure on their Borel $\sigma$-algebra. A comprehensive introduction to this theory can be found in arxiv.org/abs/2204.01125 $\endgroup$
    – Stefaan Vaes
    Commented Jun 4 at 6:05
  • $\begingroup$ Once again this is precisely what I was hoping for - thank you so much! $\endgroup$
    – szantag
    Commented Jun 4 at 10:10
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For compact (Hausdorff) topological groups I believe this is the Tannaka-Krein duality theorem, as formulated e.g. in Section II.3 of the book The Structure of Lie Groups by G. P. Hochschild (Holden-Day, 1965), see Theorem II.3.5, pp. 30. In another direction, if the group is Abelian, this just the Pontryagin duality theorem, see e.g. Theorem 24.8, pp. 378-380 of the book by E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I (2nd. edition, Springer-Verlag, 1979). In the full generality you seek, this is the whole subject of the book Kac Algebras and Duality of Locally Compact Groups By M. Enock and J.-M. Schwartz (Springer-Verlag, 1992).

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