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.
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