  In this paper, we give an alternative approach to the theory of locally
compact quantum groups, as developed by Kustermans and Vaes. We develop the
theory completely within the von Neumann algebra framework. At various points,
we also do things differently. We have a different treatment of the antipode.
We obtain the uniqueness of the Haar weights in an early stage. We take
advantage of this fact when deriving the other main results in the theory. We
also give a slightly different approach to duality. Finally, we collect, in a
systematic way, several important formulas. In an appendix, we indicate very
briefly how the $C^*$-approach and the von Neumann algebra approach eventually
yield the same objects.
