2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78433In 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.15 pages, latex 2e, amsart styleQuantum AlgebraOperator AlgebrasRings and AlgebrasRepresentation Theorytext