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.



Fourier transform on locally compact quantum groups
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...)


