2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100892We define and study an equivariant $E$-theory with respect to coactions of Hopf $C^*$-algebras; we prove the Baaj-Skandalis duality in this setting. We show that the corresponding equivariant $KK$-theory of Baaj and Skandalis enjoys an universal property. In the appendix, we look at the different ways of expressing equivariant stability for a functor, and prove an equivariant Brown-Green-Rieffel stabilization result.35 pagesK-Theory and HomologyOperator Algebras46L80; 19K56; 22D25text