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 L 1 ( 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.