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Abstract

We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if $\mathcal A$ has a brai (blai), then the right (left) module action of $\mathcal A$ on ${\mathcal A}^{*}$ is Arens regular if and only if ${\mathcal A}$ is reflexive. We find that Arens regularity is implied by the factorization of $\mathcal A^*$ or $\mathcal A^{**}$ when $\mathcal A$ is a left or a right ideal in $\mathcal A^{**}$. The Arens regularity and strong irregularity of $\mathcal A$ are related to those of the module actions of $\mathcal A$ on the $n$th dual $\mathcal A^{(n)}$ of $\mathcal A$. Banach algebras $\mathcal A$ for which $Z(\mathcal A^{**})=\mathcal A$ but $\mathcal A\subsetneq Z^t(\mathcal A^{**})$ are found (here $Z(\mathcal A^{**})$ and $Z^t(\mathcal A^{**})$ are the topological centres of $\mathcal A^{**}$ with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that $\mathcal A\subsetneq Z(\mathcal A^{**})\subsetneq \mathcal A^{**}.$ Finally, the triangular Banach algebras $\mathcal T$ are used to find Banach algebras having the following properties: (i) ${\mathcal T}^{*}{\mathcal T} = {\mathcal T} {\mathcal T}^{*}$ but $Z({\mathcal T}^{**})\ne Z^t({\mathcal T}^{**})$; (ii) $Z({\mathcal T}^{**}) = Z^t({\mathcal T}^{**})$ and ${\mathcal T}^{*}{\mathcal T} = {\mathcal T}^{*}$ but ${\mathcal T} {\mathcal T}^{*} \ne {\mathcal T}^{*}$; (iii) $Z(\mathcal T^{**})=\mathcal T$ but $\mathcal T$ is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and Ülger.