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In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure analog). The deficiency is that they had to presuppose existence when one would expect to derive it. The existence can be proven for compact quantum groups, where a multiplicative unit is assumed.

My question: What is the current status on this? Has the existence of the Haar measure analog been derived or is it still an axiom? The papers are not that recent, so maybe they achieved it.

A bonus question: What about Haar systems for locally compact quantum groupoids? are they an axiom?

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    $\begingroup$ I'm only a spectator of LCQG stuff, but my impression is that this is still open. $\endgroup$ – Yemon Choi Sep 23 '14 at 18:05
  • $\begingroup$ If is necessarily an axiom, what then? What would it mean? A revision of the topological Hopf structures in order to derive the existence would be needed? Or only "Haar Quantum groups" should be studied? I'm sorry for being so demanding. $\endgroup$ – Henrique Tyrrell Sep 23 '14 at 19:10
  • $\begingroup$ Henrique, I'm afraid this is where I'd have to defer to an expert, but there are some people who would know and who are occasionally on MO $\endgroup$ – Yemon Choi Sep 23 '14 at 20:41
  • $\begingroup$ So let us hope this question shows up to them. $\endgroup$ – Henrique Tyrrell Sep 23 '14 at 21:56
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    $\begingroup$ As to my knowledge, it is open, and there is almost no activity in solving it. $\endgroup$ – Yulia Kuznetsova Oct 6 '14 at 10:34

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