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Associated with a locally compact group $\cal G$ and a $\cal G$-space $\calX$ there is a Banach subspace $LUC({\cal X},{\cal G})$ of $C_b({\cal X})$,which has been introduced and studied by Lau and Chu in \cite{chulau}. In thispaper, we study some properties of the first dual space of $LUC({\cal X},{\calG})$. In particular, we introduce a left action of $LUC({\cal G})^*$ on$LUC({\cal X},{\cal G})^*$ to make it a Banach left module and then weinvestigate the Banach subalgebra ${{\frak{Z}}({\cal X},{\cal G})}$ of$LUC({\cal G})^*$, as the topological centre related to this module action,which contains $M({\cal G})$ as a closed subalgebra. Also, we show that thefaithfulness of this module action is related to the properties of the actionof $\cal G$ on $\cal X$ and we extend the main results of Lau~\cite{lau} fromlocally compact groups to ${\cal G}$-spaces. Sufficient and/or necessaryconditions for the equality ${{\frak{Z}}({\cal X},{\cal G})}=M({\cal G})$ or$LUC({\cal G})^*$ are given. Finally, we apply our results to some specialcases of $\cal G$ and $\cal X$ for obtaining various examples whose topologicalcentres ${{\frak{Z}}({\cal X},{\cal G})}$ are $M({\cal G})$, $LUC({\cal G})^*$or neither of them.

On the weak$^*$ continuity of $LUC({\cal G})^*$-module action on $LUC({\cal X},{\cal G})^*$ related to $\cal G$-space $\cal X$

On the weak$^$ continuity of $LUC({\cal G})^$-module action on $LUC({\cal X},{\cal G})^*$ related to $\cal G$-space $\cal X$