I am trying to understand what seems to be a common knowledge, that every compact quantum group has a Haar state. However, each approach that I found on the internet is a bit hard for me to grasp.
I began with Van Daele's paper: http://www.ams.org/journals/proc/1995-123-10/S0002-9939-1995-1277138-0/S0002-9939-1995-1277138-0.pdf
However, I do not understand, why the weak$^*$ limit in Lemma 2.1 exists. In other words, given a state $\omega$, why does a sequence of Cesaro sums $$\omega_n = \frac{1}{n} \bigg(\omega + \omega^{\star 2} +\ldots + \omega^{\star n}\bigg)$$
converge in the weak sense? $\omega^{\star n}$ denotes the $n$-convolution, i.e. $$\phi\star\psi = (\phi\otimes\psi) \circ \Delta, \hspace{0.4cm} \Delta - \text{comultiplication}.$$
In Woronowicz's paper: https://www.impan.pl/~pmh/teach/intro3/CQG3.pdf
the issue is omitted but the price we have to pay is that we assume the compact quantum group to be separable. I would like not to do that, although I heard that non-separable $C^*$-algebras are considered as 'pathological'. Is there any evidence to support that claim?
Furthermore, I also stumbled upon a paper 'Haar Measures on Hopf $C^*$-Algebras' by Quan (I don't know if enclosing the link would be legal). The idea of using Markov-Kakutani fixed point theorem is really breathtaking (in my opinion), but still, the author needs to assume that the compact quantum group is cocommutative. I would also like to avoid this restriction.
Last but not least, there is another Woronowicz's paper: https://projecteuclid.org/download/pdf_1/euclid.cmp/1104159726
which concerns compact matrix pseudogroups (I don't know how these relate to compact quantum groups yet). The exposition is hard for me to follow, but I suspect the answer to my question may be somewhere between the lines of Lemma 2.6 and Proposition 2.7.
To sum up, I would like to find out a 'relatively modern approach' (not in the language of matrix pseudogroups) to the construction of Haar state on a compact quantum group, which is neither separable nor cocommutative.
Every bounded ball in the dual space is $w^{*}$ compact by the Banach-Alaoglu theorem, in particular, every bounded net has a $w^{*}$-convergent subnet. The state $\omega_n$ is bounded as an average of states, and so it has a convergent subnet in the $w^{*}$ topology, which implies that the sequence has a limit point in the $w^{*}$ topology. I don't think you can assert that the sequence converges itself, though.