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...)

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...)