  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.
