I Have read some articles on locally compact quantum groups and the Fourier transform on them. I wonder yet 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 $L^\infty(Gr({\mathbb{G}}))$ 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 $Gr(L^\infty(G))=\widehat{G}= sp(L^1(G))$ and $\mathcal{F}:L^1(G)\rightarrow L^\infty(\widehat{G})$.

I really appreciate if anybody help me in this regard.