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The purely algebraic notion of CQG algebra (algebra of functions on a compactquantum group) is defined. In a straightforward algebraic manner, thePeter-Weyl theorem for CQG algebras and the existence of a unique positivedefinite Haar functional on any CQG algebra are established. It is shown that aCQG algebra can be naturally completed to a $C^\ast$-algebra. The relationsbetween our approach and several other approaches to compact quantum groups arediscussed.Comment: 14 pp., Plain TeX, accepted by Lett. Math. Phy

CQG algebras: A direct algebraic approach to compact quantum groups