Measured quantum groupoids on a finite basis and equivariant Kasparov theory

In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning the equivariant Kasparov theory for actions of locally compact quantum groups, see Baaj and Skandalis (1989, 1993). To every pair (A, B) of C*-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis G, we associate a G-equivariant Kasparov theory group KKG(A, B). The Kasparov product generalizes to this setting. By applying recent results by Baaj and Crespo (2017, 2018) concerning actions of regular measured quantum groupoids on a finite basis, we obtain two canonical homomorphisms J(G) : KKG(A, B) -> KK(G) over cap(A (sic) G, B (sic) G) and J((G) over cap ): KKG(A , B) -> KKG(A (sic) (G) over cap, B (sic) (G) over cap) inverse of each other through the Morita equivalence coming from a version of the Takesaki-Takai duality theorem. We investigate in detail the case of colinking measured quantum groupoids. In particular, if G(1) and G(2) are two monoidally equivalent regular locally compact quantum groups, we obtain a new proof of the canonical equivalence of the associated equivariant Kasparov categories, see Baaj and Crespo (2017).