Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$.
What is a precise description of a maximal ,or in some sense widest as possible, category of compact topological groups $\mathcal{TG}$ for which the following functor is well defined:
$$\mathcal{F}:\mathcal{FG}\to \mathcal{A} \\ \mathcal{F}(G)=C(\tilde{G}) $$ where $C(\tilde{G})$ is the algebra of all continuous functions on Haussdorf space $\tilde{G}$ where the latter is equiped with the quotient topology imposed by the natural quotient map $G\to \tilde{G}$
Now how can this functor be naturally and maximally extended to an appropriate category of quantum groups?
