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 LL can be identified with MG. As a consequence, we prove that G is compact if and only if LUC={ rm WAP}( G) and MG cong mathcal{Z}({ rm LUC}( G)^*) ; which partially answer a problem raised by Volker Runde.

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