Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A∗, and show that all A-module homomorphisms of A∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L1(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC∗(G)); which partially answer a problem raised by Volker Runde.