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

Research paper by H. Javanshiri, N. Tavallaei

Indexed on: 15 Jun '18Published on: 14 Jun '18Published in: Semigroup Forum

#### 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.