  In this paper we construct and study the representation theory of a Hopf
C^*-algebra with approximate unit, which constitutes quantum analogue of a
compact group C^*-algebra. The construction is done by first introducing a
convolution-product on an arbitrary Hopf algebra H with integral, and then
constructing the L_2 and C^*-envelopes of H (with the new convolution-product)
when H is a compact Hopf *-algebra.
