Suppose that a type $II_{1}$ factor $M$ decomposes in two ways as a group von Neumann algebra, e.g. as $L\Gamma$ and as $L\Lambda$. The decomposition $L\Gamma$ gives rise to a comultiplication $$\Delta_{\Gamma}:M\rightarrow M\overline{\otimes}M$$ that sends a canonical unitary $u_g$ to $u_{g}\otimes u_{g}$. There is an analogous comultiplication $\Delta_{\Lambda}$. On page 8 of Stefaan Vaes's notes, it is shown that if $\Omega$ is a unitary element in $M\overline{\otimes}M$ such that $$\Delta_{\Gamma}(x)=\Omega \Delta_{\Lambda}(x)\Omega^{*}$$ for all $x\in M$, then a certain set of equations hold which, if the group $\Gamma$ were instead abelian, would yield a symmetric 2-cocycle on the dual compact abelian group. Such cocycles cobound, and this fact leads Ioana, Popa and Vaes to prove an analogous fact in the nonabelian setting.

To obtain the set of equations in question, one writes down an element $$Z=(\Delta_{\Gamma}\otimes id)(\Omega)(\Omega \otimes 1)(1 \otimes \Omega^{*})(id \otimes \Delta_{\Gamma})(\Omega^{*})$$ which is unitary because of coassociativity properties of the two comultiplications. The set of equations referred to in the previous paragraph follow because $M$ is a factor.

Question:Is the consideration of this unitary $Z$ completely ad-hoc, or is there some deeper reason for considering an element of this form that is perhaps related to thefundamental unitarythat encodes duality in the theory of locally compact quantum groups?

I think the idea of developing the link between the quantum group picture and $II_{1}$ factors is intriguing, and so posted this question instead of sending S. Vaes an e-mail, in hope that an expert would provide an answer for the OA community.