  Consider a C*-algebra $A$ with a comultiplication $\Delta$. This pair is
usually thought of as locally compact quantum semi-group. When these notes were
written, in 1993, it was not at all clear what the extra assumptions on the
comultiplication should be for this pair to be a 'locally compact quantum
group'. This only became clear in 1999 thanks to the work of Kustermans and
Vaes. In the compact case however, rather natural conditions are formulated by
Woronowicz and a good notion of a compact quantum group was available in 1993.
  In these notes, we consider another class of locally compact quantum groups.
We assume the existence of a non-zero element $h$ in $A$ satisfying
$\Delta(a)(1\otimes h)=a \otimes h$ for all $a$ in $A$. We add some natural
conditions and we speak of a 'quasi-discrete locally compact quantum group'. We
also discuss the discrete case. We prove the existence of the Haar measure, the
regular representation, the fundamental unitary and we obtain the reduced dual.
  The quasi-discrete case is, in principle, more general than the discrete
case. Shortly after these notes were written, Kustermans showed that in fact,
the two classes coincided.
  For this reason, these notes have not been published. Nevertheless, some of
the results and techniques seem to be usful and in recent work, we came across
similar settings. Therefore, we have decided to make these notes available on
the net. We have added some comments at the end of the introduction and we have
updated the reference list. But apart from these minor changes, the notes are
still as they were written and distributed in 1993.
