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.