Module maps over locally compact quantum groups

Volume 211 / 2012

Zhiguo Hu, Matthias Neufang, Zhong-Jin Ruan Studia Mathematica 211 (2012), 111-145 MSC: 22D15, 43A22, 43A30, 46H05. DOI: 10.4064/sm211-2-2


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.


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

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