Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$. 

It is well-known that we can make the following construction:
- Embed $A\subseteq B(L^2(A, \psi)).$
- Define $M= A''\subseteq B(L^2(A, \psi))$.
- Show that $M$ with an extension of the coproduct $\Delta$ becomes a von Neumann algebraic locally compact quantum group in the sense of Kustermans and Vaes.

See for instance Van Daele's recent survey paper [Algebraic quantum groups and duality II][1] for details about this construction.

In this case, we can show that the right invariant nsf weight $\widetilde{\psi}$ extends $\psi$. I'm interested in the following: can we show (using elementary means) that a right-invariant nsf weight $\widetilde{\psi}: M_+\to [0, \infty]$ extending $\psi: A\cap M^+  \to \mathbb{C}$ is necessarily unique? Obviously, this is true since in the general theory of locally compact quantum groups since a right invariant nsf weight is uniquely determined (up to constant). I am interested to see a more elementary approach in this case though.

Note that if $A$ is discrete, this is fairly easy, since a positive element in $M$ is a $\sigma$-strong limit of an increasing net of positive elements in $A$, so normality of the weight does the trick. 



  [1]: https://arxiv.org/pdf/2304.13482.pdf