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I have read some articles on locally compact quantum groups and the Fourier Transform on them. I wonder why we define the Fourier transform as an operator valued functions from $L^1(\mathbb{G})$ to $L^\infty(\widehat{\mathbb{G}})$.

Why do not we used the intrinsic group $Gr(\mathbb{G})$ which has been defined by Mehrdad Kalantar, and define the Fourier transform from $L^1(\mathbb{G})$ to $L^\infty(\operatorname{Gr}({\mathbb{G}}))$ as a complex-valued function? I think if we do it we can see immediately that it is an analogue of the Fourier transform in the classical case, since when we work with a locally compact Abelian group $G$ we know that $\operatorname{Gr}(L^\infty(G))=\widehat{G}= \operatorname{sp}(L^1(G))$ and $\mathcal{F}:L^1(G)\rightarrow L^\infty(\widehat{G})$.

I would really appreciate it if anybody could help me in this regard.

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    $\begingroup$ The intrinsic group forgets information, so your candidate does not have various features one would desire of a Fourier transform. I would answer your question "Why not?" with "Why should we?" $\endgroup$
    – Yemon Choi
    Commented May 31, 2012 at 5:48
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    $\begingroup$ I was also not aware that the intrinsic group was first defined by Kalantar, but I am not an expert in the literature. $\endgroup$
    – Yemon Choi
    Commented May 31, 2012 at 5:49
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    $\begingroup$ To adjust my previous statement: after some informal consultation, it seems increasingly likely that the concept of intrinsic group predates Kalantar's work. $\endgroup$
    – Yemon Choi
    Commented Jun 6, 2012 at 8:53

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My view is that one should treat LCQGs as a self-dual category; so there is no reason to prejudice, for a classical group $G$, the commutative case (leading to $L^1(\G)$) over the co-commutative (leading to $A(G)$).

The co-commutative is nice from the point of view of intrinsic groups-- this goes back to Takesaki and Tatsumma (and arguably Eymard, Herz etc.) where they showed that the intrinsic group of $VN(G)$ is just $G$ (with the same topology).

But in the commutative case, it's awful-- the intrinsic group of $L^1(G)$ is just the group of characters of $G$, which is rarely interesting outside of the abelian group case. Well, "interesting" is a bit extreme, giving maximal tori etc., but it certainly wouldn't give an injective Fourier transform.

(I think here maybe I have computed things in the "dual" formalism to that of the original question).

For a quantum example, I think Mehrdad showed that for $SU_\mu(2)$, you just get the maximal torus; so again the Fourier transform fails to be injective. That's not going to lead to an interesting theory (unless you have some specific application already in mind...)

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  • $\begingroup$ Dear Prof. Daws, Thank you. As we have nice analogues of lots of Abstract Harmonic Analysis's concepts in the locally compact quantum group setting, I was thinking that there should be a quantum group analogue of Wienner Tauberian theorem (In the first chapter of Fourier Analysis of Rudin) as well. So this question led me to define Fourier Transform like this. $\endgroup$
    – Zora
    Commented Jun 4, 2012 at 1:32
  • $\begingroup$ Zora, generalizing results for commutative Banach algebras to noncommutative ones can be quite subtle, or just plain impossible. It is (IMHO) occasionally advisable to try and understand certain objects one defines before trying to prove hasty results about them $\endgroup$
    – Yemon Choi
    Commented Jun 6, 2012 at 8:51
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This paper of van Daele seems to be pretty convincing as to the naturality of Fourier transform (it also is pretty pleasing in avoiding the analysis...)

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    $\begingroup$ I have read van Daele's paper and even papers of Martin Casper and Kahng on Fourier transform on locallay compact quantum groups. But I still do not understand Why do not we used the intrinsic group Gr(G) which has been defined by Mehrdad Kalantar and define the Fourier transform from L1(G) to L∞(Gr(G)) as a complex-valued function. I think if we do it we can see immediately that it is an analogue of Fourier transform in classical case, since when we work with locally compact Abelian group G we know that Gr(L∞(G))=Gˆ=sp(L1(G)) and F:L1(G)→L∞(Gˆ). $\endgroup$
    – Zora
    Commented May 31, 2012 at 2:15

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