TITLE: What is the discrete quantum group associated to a compact group?
QUESTION [10 upvotes]: Let $G$ be a compact topological group. Then $G$ is a CQG with function algebra $C(G)$ and the usual comultiplication on $C(G)$. Is there an easy description of the dual discrete quantum group $\widehat{G}$?
In the case of commutative compact groups, I would hope that there is a connection with the usual dual of a compact group (in the sense of Pontryagin duality), but since the function algebra of the discrete quantum group $\widehat{G}$ is a von Neumann algebra, it won't be something nice like $C(\widehat{G})$.

REPLY [4 votes]: It's not my area, but I think the answer is very simple: when $G$ is compact abelian the W${}^*$ dual quantum group is just $L(G) \cong l^\infty(\hat{G})$.