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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 the fundamental unitary that 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.

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as someone who knows very little about II-1 factors but knows a bit more about cohomology, I think this is a great question and would also like to know more... –  Yemon Choi Nov 29 '12 at 0:06
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1 Answer

Perhaps you can take a look at Chapter 3 of CQG and their rep cats? A good part of it makes sense for locally compact quantum groups.

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Thanks, Makoto! I'll have a look. –  Jon Bannon Nov 30 '12 at 22:12
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@Jon: note that when $\Gamma$ is any discrete group then $L\Gamma$ with standard comultiplication is a compact quantum group in the sense of those notes (being the "algebra of functions" on some "compact noncommutative spaces) - so one doesn't need the L in LCQG –  Yemon Choi Dec 1 '12 at 1:48
    
It does seem that Ch. 3 may have something to say, here. Also, thanks for the supplementary comment, Yemon. –  Jon Bannon Dec 1 '12 at 1:50
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