Arens regularity of module actions
Studia Mathematica, Tome 181 (2007) no. 3, pp. 237-254 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/sm181-3-3
Keywords: study arens regularity module actions banach right modules banach algebras prove mathcal has brai blai right module action mathcal mathcal * arens regular only mathcal reflexive arens regularity implied factorization mathcal * mathcal ** mathcal right ideal mathcal ** arens regularity strong irregularity mathcal related those module actions mathcal nth dual mathcal mathcal banach algebras mathcal which mathcal ** mathcal mathcal subsetneq mathcal ** found here mathcal ** mathcal ** topological centres mathcal ** respect first second arens product respectively gives examples banach algebras mathcal subsetneq mathcal ** subsetneq mathcal ** finally triangular banach algebras mathcal banach algebras having following properties nbsp mathcal * mathcal mathcal mathcal * mathcal ** mathcal ** nbsp mathcal ** mathcal ** mathcal * mathcal mathcal * mathcal mathcal * mathcal * iii nbsp mathcal ** mathcal mathcal weakly sequentially complete results iii examples answering questions asked lau lger

M. Eshaghi Gordji  1   ; M. Filali  2

1 Department of Mathematics University of Semnan Semnan, Iran and Department of Mathematics Shahid Beheshti University Tehran, Iran
2 Department of Mathematical Sciences University of Oulu Oulu 90014, Finland
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M. Eshaghi Gordji; M. Filali. Arens regularity of module actions. Studia Mathematica, Tome 181 (2007) no. 3, pp. 237-254. doi: 10.4064/sm181-3-3

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