Actions of measured quantum groupoids on a finite basis

Abstract

In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C$^{*}$-algebras (Comm. Math. Phys. 235 (2003) 139–167). Let $\mathcal{G}$ be a measured quantum groupoid on a finite basis. We prove that if $\mathcal{G}$ is regular, then any weakly continuous action of $\mathcal{G}$ on a C$^{*}$-algebra is necessarily strongly continuous. Following (K-Theory 2 (1989) 683–721), we introduce and investigate a notion of $\mathcal{G}$-equivariant Hilbert C$^{*}$-modules. By applying the previous results and a version of the Takesaki–Takai duality theorem obtained in (Bull. Soc. Math. France 145 (2017) 711–802) for actions of $\mathcal{G}$, we obtain a canonical equivariant Morita equivalence between a given $\mathcal{G}$-C$^{*}$-algebra $A$ and the double crossed product $(A\rtimes\mathcal{G})\rtimes\widehat{\mathcal{G}}$.