When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) and represent it as (unbounded) operators on some Hilbert space. Then, we generate (say in the von Neumann algebraic sense) the algebra $M$ of the LCQG from the closures of the 'coordinate' operators, i.e. $M := W^*(\{\bar{x}^i\})$. (Let $\bar{A}_q$ denote the *-subalgebra of unbounded operators generated (in the *-algebraic sense) by the closed $\bar{x}^i$).
Intuitively, we want not only the 'coordinate' functions, but the whole $A_q$ generated by them to be (almost) included in $M$. In short, my question is: Is it $A_q$ that is affiliated with $M$ or rather the $*$-algebra $\bar{A}_q$ of polynomials in closures of the 'coordinates'?
Classically, both of them work. Quantised, I am not sure. Their pros/cons I see:
$A_q$ is represented inside of $\mathcal{L}^+(D)$, i.e. the *-algebra of the adjoinable linear operators in $\text{End }D$ for some dense $D$. This is nice as it guarantees that all the operators in $A_q$ are closable. My problem is that, if I am not mistaken, the non-closed 'coordinates' $x^i$ are not necessarily affiliated with $M$...
all elements of $\bar{A}_q$ are automatically affiliated with $M$ and so are their closures. The problem, if I am also not mistaken, is that elements of $\bar{A}_q$ are not necessarily closable (i.e. the product of two closable operators can be non-closable?), which is obviously a desirable trait.
EDIT: I realised that of course all elements of $\bar{A}_q$ are closable so it indeed is a well-defined candidate for an 'algebra of coordinates'. So my only question would be whether $A_q$ couls also work, i.e are $x^i$ always affiliated with $M$?
(Above I omitted the issue with symmetric elements of these algebras and their essential self-adjointness. The whole post is inspired by reading about $\widetilde{SU}_q(1, 1)$ (https://arxiv.org/abs/math/0105117) but I believe the question to be more about the spirit of the 'algorithm' of operator algebraic deformation of matrix groups)
