2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64205In this paper, we study the duality theory of Hopf $C^*$-algebras in a general ``Hilbert-space-free'' framework. Our particular interests are the ``full duality'' and the ``reduced duality''. In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf $C^*$-algebra.It is a shortened version of a paper with the same title that was in circulation since 1998. To be published in the International Journal of MathematicsOperator AlgebrasQuantum Algebra46L55; 46L89text