The canonical central exact sequence for locally compact quantum groups

For a locally compact quantum group G we define its center, Z(G), and its quantum group of inner automorphisms, Inn (G). We show that one obtains a natural isomorphism between Inn (G) and G/Z(G), we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.