Measured quantum groupoids on a finite basis and equivariant Kasparov theory

Abstract

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 ${\mathrm{C}}^{\ast }$-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis $\mathcal{G}$, we associate a $\mathcal{G}$-equivariant Kasparov theory group ${\mathsf{KK}}_{\mathcal{G}}(A,B)$. 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 $J_{\mathcal{G}}:{\mathsf{KK}}_{\mathcal{G}}(A,B)\to {\mathsf{KK}}_{\hat{\mathcal{G}}}(A⋊\mathcal{G},B⋊\mathcal{G})$ and $J_{\hat{\mathcal{G}}}:{\mathsf{KK}}_{\hat{\mathcal{G}}}(A,B)\to {\mathsf{KK}}_{\mathcal{G}}(A⋊\hat{\mathcal{G}},B⋊\hat{\mathcal{G}})$ 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 ${\mathbb{G}}_1$ and ${\mathbb{G}}_2$ 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).