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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Module maps over locally compact quantum groups

### Tom 211 / 2012

Studia Mathematica 211 (2012), 111-145 MSC: 22D15, 43A22, 43A30, 46H05. DOI: 10.4064/sm211-2-2

#### Streszczenie

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.

#### Autorzy

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

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