Comptes Rendus de l'Académie des Sciences. Série I, Mathématique vol:328 issue:10 pages:871-876
In this Note 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, and hence we define in fact arbitrary locally compact quantum groups represented on the L-2-space of the Haar weight. We construct the multiplicative unitary associated with our quantum group. We construct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontryagin duality theorem. (C) Academie des Sciences/Elsevier, Paris.