The analytic structure of algebraic quantum groups

In [Adv. Math. 140 (1998) 323–366], Van Daele introduced the notion of an algebraic quantum group. We proved in [Internat. J. Math. 8 (8) (1997) 1067–1139] that such algebraic quantum groups give rise to reduced C*-algebraic quantum groups in the sense of [J. Kustermans, S. Vaes, Ann. Sci. École Norm. Sup. (4) 33 (2000) 837–934]. After introducing the universal C*-algebraic quantum group associated to an algebraic one, we will pull down the analytic structure of this C*-algebraic quantum group to the algebraic quantum group. The multiplier algebra of the dual quantum group will be realized as a space of linear functionals on the original algebra. We also identify the analytic structure of the dual quantum group in terms of the analytic structure of the original algebraic quantum group.