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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.