Some properties of $LUC({\mathcal X},{\mathcal G})^*$ as a banach left $LUC({\mathcal G})^*$-module

Abstract

Associated with a locally compact group $\mathcal G$ and a $\mathcal G$-space $\mathcal X$ there is a Banach subspace $LUC({\mathcal X},{\mathcal G})$ of $C_b({\mathcal X})$, which has been introduced and studied by Chu and Lau (Math Z 268:649–673, 2011). In this paper, we study some properties of the first dual space of $LUC({\mathcal X},{\mathcal G})$. In particular, we introduce a left action of $LUC({\mathcal G})^*$ on $LUC({\mathcal X},{\mathcal G})^*$ to make it a Banach left module and then we investigate the Banach subalgebra ${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}$ of $LUC({\mathcal G})^*$, as the topological centre related to this module action, which contains $M({\mathcal G})$ as a closed subalgebra. Also, we show that the faithfulness of this module action is related to the properties of the action of $\mathcal G$ on $\mathcal X$ and we prove an analogue of the main result of Lau (Math Proc Cambridge Philos Soc 99:273–283, 1986) for ${\mathcal G}$-spaces. Sufficient and/or necessary conditions for the equality ${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}=M({\mathcal G})$ or $LUC({\mathcal G})^*$ are given. Finally, we apply our results to some special cases of $\mathcal G$ and $\mathcal X$ for obtaining various examples whose topological centres ${{\mathfrak {Z}}({\mathcal X},{\mathcal G})}$ are $M({\mathcal G})$, $LUC({\mathcal G})^*$ or neither of them.