Module maps over locally compact quantum groups

We study locally compact quantum groups 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 L1(G) are used to characterize strong Arens irregularity of L _∞(G) and are linked to commutation relations over 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) ? WAP(G) to all locally compact quantum groups.