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?

2more comments