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 equivariant Kasparov theory for actions of locally compact quantum groups [S. Baaj and G. Skandalis, 1989, 1993]. To every pair $(A,B)$ of C*-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 concerning actions of regular measured quantum groupoids on a finite basis [S. Baaj and J. C., 2015; J. C., 2017], 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 [S. Baaj and J. C., 2015; J. C., 2017]. 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 [S. Baaj and J. C., 2015].

This paper is a follow-up to the article "Actions of measured quantum groupoids on a finite basis" [arXiv:1706.08292] from the same author. The introductory chapters are essentially identical so that the present article can almost be read independently