\'Equivalence mono\"idale de groupes quantiques et K-th\'eorie bivariante

In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_1$ and $G_2$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories ${A}^{G_1}$ and ${A}^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on $C^*$-algebras. As an application of this result, we derive a canonical equivalence of the categories ${KK}^{G_1}$ and ${KK}^{G_2}$. We introduce and investigate a notion of actions on $C^*$-algebras of measured quantum groupoids on a finite basis. The proof of the equivalence between ${KK}^{G_1}$ and ${KK}^{G_2}$ relies on a version of the Takesaki-Takai duality theorem for continuous actions on $C^*$-algebras of measured quantum groupoids on a finite basis.