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

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 Lau and Chu in \cite{chulau}. 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 $LUC(\mathcal{G}{)}^{\ast }$ on $LUC(\mathcal{X},\mathcal{G}{)}^{\ast }$ to make it a Banach left module and then we investigate the Banach subalgebra $\mathfrak{Z}(\mathcal{X},\mathcal{G})$ of $LUC(\mathcal{G}{)}^{\ast }$, 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 extend the main results of Lau~\cite{lau} from locally compact groups to $\mathcal{G}$-spaces. Sufficient and/or necessary conditions for the equality $\mathfrak{Z}(\mathcal{X},\mathcal{G})=M(\mathcal{G})$ or $LUC(\mathcal{G}{)}^{\ast }$ 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 $M(\mathcal{G})$, $LUC(\mathcal{G}{)}^{\ast }$ or neither of them.