  The purely algebraic notion of CQG algebra (algebra of functions on a compact
quantum group) is defined. In a straightforward algebraic manner, the
Peter-Weyl theorem for CQG algebras and the existence of a unique positive
definite Haar functional on any CQG algebra are established. It is shown that a
CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations
between our approach and several other approaches to compact quantum groups are
discussed.
