Module maps over locally compact quantum groups

Zhiguo Hu ¹   ; Matthias Neufang ²   ; Zhong-Jin Ruan ³

¹ Department of Mathematics and Statistics University of Windsor Windsor, Ontario, Canada N9B 3P4
² School of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada K1S 5B6 and Université Lille 1 – Sciences et Technologies UFR de Mathématiques Laboratoire de Mathématiques Paul Painlevé UMR CNRS 8524 59655 Villeneuve d'Ascq Cédex, France
³ Department of Mathematics University of Illinois Urbana, IL 61801, U.S.A.

Studia Mathematica, Tome 211 (2012) no. 2, pp. 111-145

Résumé

We study locally compact quantum groups $\mathbb{G}$ and their module maps through a general Banach algebra approach. As applications, we obtain various characterizations of compactness and discreteness, which in particular generalize a result by Lau (1978) and recover another one by Runde (2008). Properties of module maps on $L_\infty(\mathbb{G})$ are used to characterize strong Arens irregularity of $L_1(\mathbb{G})$ and are linked to commutation relations over $\mathbb{G}$ with several double commutant theorems established. We prove the quantum group version of the theorems by Young (1973), Lau (1981), and Forrest (1991) regarding Arens regularity of the group algebra $L_1(G)$ and the Fourier algebra $A(G)$. We extend the classical Eberlein theorem on the inclusion $B(G) \subseteq \mathit{WAP} (G)$ to all locally compact quantum groups.

Détail

DOI : 10.4064/sm211-2-2

Keywords: study locally compact quantum groups mathbb their module maps through general banach algebra approach applications obtain various characterizations compactness discreteness which particular generalize result lau recover another runde properties module maps infty mathbb characterize strong arens irregularity mathbb linked commutation relations mathbb several double commutant theorems established prove quantum group version theorems young lau forrest regarding arens regularity group algebra fourier algebra extend classical eberlein theorem inclusion subseteq mathit wap locally compact quantum groups