In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually make sense to develop the theory of nonseparable compact quantum groups? What has gone wrong?
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6$\begingroup$ Carlo has given a reference to a general framework by van Daele, but it is worth noting that the reduced ${\rm C}^*$-algebra of any discrete group has always been recognized as an example of a compact quantum group (in fact, a compact Kac algebra) and this will be non-separable as soon as the discrete group is uncountable $\endgroup$– Yemon ChoiCommented Sep 15, 2018 at 12:42
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A Haar measure on a compact quantum group without requiring separability was constructed in The Haar measure on a compact quantum group (1995).
answered Sep 15, 2018 at 12:37