We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if has a brai (blai), then the right (left) module action of on * is Arens regular if and only if is reflexive. We find that Arens regularity is implied by the factorization of * or ** when is a left or a right ideal in **. The Arens regularity and strong irregularity of are related to those of the module actions of on the nth dual of . Banach algebras for which Z( **) = but are found (here Z( **) and are the topological centres of ** with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that ⊊ Z( **) ⊊ **. Finally, the triangular Banach algebras are used to find Banach algebras having the following properties: (i) * = * but ; (ii) and * = * but * ≠ *; (iii) Z(**) = but is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and Ülger.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-3-3,
author = {M. Eshaghi Gordji and M. Filali},
title = {Arens regularity of module actions},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {237-254},
zbl = {1165.46024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-3-3}
}
M. Eshaghi Gordji; M. Filali. Arens regularity of module actions. Studia Mathematica, Tome 178 (2007) pp. 237-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-3-3/