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

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.