Open Access
2018 Actions of measured quantum groupoids on a finite basis
Jonathan Crespo
Illinois J. Math. 62(1-4): 113-214 (2018). DOI: 10.1215/ijm/1552442659

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 $(A\rtimes\mathcal{G})\rtimes\widehat{\mathcal{G}}$.

Citation

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Jonathan Crespo. "Actions of measured quantum groupoids on a finite basis." Illinois J. Math. 62 (1-4) 113 - 214, 2018. https://doi.org/10.1215/ijm/1552442659

Information

Received: 6 November 2017; Revised: 23 August 2018; Published: 2018
First available in Project Euclid: 13 March 2019

zbMATH: 07036783
MathSciNet: MR3922413
Digital Object Identifier: 10.1215/ijm/1552442659

Subjects:
Primary: 16T99 , 46L55 , 46L89

Rights: Copyright © 2018 University of Illinois at Urbana-Champaign

Vol.62 • No. 1-4 • 2018
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