Module maps over locally compact quantum groups
Volume 211 / 2012
Abstract
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.
