Locally compact quantum groups

In this paper we propose a simple definition of a locally compact quantum group in reduced form. By the word 'reduced' we mean that we suppose the Haar weight to be faithful. So in fact we define and study an arbitrary locally compact quantum group, represented on the L-2-space of its Haar weight. For this locally compact quantum group we construct the antipode with polar decomposition. We construct the associated multiplicative unitary and prove that it is manageable in the sense of Woronowicz. We define the modular element and prove the uniqueness of the Haar weights. Following [15] we construct the reduced dual, which will again be a reduced locally compact quantum group. Finally we prove that the second dual is canonically isomorphic to the original reduced locally compact quantum group, extending the Pontryagin duality theorem. (C) 2000 Editions scientifiques et medicales Elsevier SAS.