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 $A^{\ast }$, and show that all $A$-module homomorphisms of $A^{\ast }$ are normal if and only if $A$ is an ideal of $A^{\ast \ast }$. We obtain some characterizations of compactness and discreteness for a locally compact quantum group $\text{G}$. Furthermore, in the co-amenable case we prove that the multiplier algebra of $\text{LL}$ can be identified with $\text{MG}\mathrm{.}$ As a consequence, we prove that $\text{G}$ is compact if and only if $\text{LUC}=\mathrm{W}\mathrm{A}\mathrm{P}(\text{G})$ and $\text{MG}\cong \mathcal{Z}(\mathrm{L}\mathrm{U}\mathrm{C}(\text{G}{)}^{\ast })$; which partially answer a problem raised by Volker Runde.