Associated with a locally compact group $\cal G$ and a $\cal G$-space $\cal X$ 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 this paper, we study some properties of the first dual space of $LUC({ \cal X},{ \cal G})$. 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 we investigate 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 the faithfulness of this module action is related to the properties of the action of $\cal G$ on $\cal X$ and we extend the main results of Lau~\cite{ lau} from locally compact groups to ${ \cal G}$-spaces. Sufficient and/or necessary conditions 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 special cases of $\cal G$ and $\cal X$ for obtaining various examples whose topological centres ${ { \frak{ Z}}({ \cal X},{ \cal G})}$ are $M({ \cal G})$, $LUC({ \cal G})^*$ or neither of them.
Categories: math.FA