  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.
