Compact, open subgroups of a quantum group and multiplier Hopf *-algebras

(Joint work with A. Van Daele) The existence of a multiplier Hopf *-algebra of functions on a locally compact group G (pointwise product) is equivalent to the existence of a compact, open subgroup. This is again equivalent to the following: There are non-zero elements $a\in C^*_r(G)$ and $b\in C_0(G)$ which commute. We shall use this to discuss how to characterize compact, open "subgroups" of a quantum group and what a totally disconnected quantum group should be.