Arens regularity of module actions

M. Eshaghi Gordji, and M. Filali. "Arens regularity of module actions." Studia Mathematica 181.3 (2007): 237-254. <http://eudml.org/doc/286315>.

@article{M2007,
abstract = {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 $^\{(n)\}$ of . Banach algebras for which Z( **) = but $ ⊊ Z^\{t\}( **)$ are found (here Z( **) and $Z^\{t\}( **)$ 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 $Z(**) ≠ Z^\{t\}(**)$; (ii) $Z(**) = Z^\{t\}(**)$ 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.},
author = {M. Eshaghi Gordji, M. Filali},
journal = {Studia Mathematica},
keywords = {Arens product; topological centres; strongly Arens irregular; module actions; triangular Banach algebras},
language = {eng},
number = {3},
pages = {237-254},
title = {Arens regularity of module actions},
url = {http://eudml.org/doc/286315},
volume = {181},
year = {2007},
}

TY - JOUR
AU - M. Eshaghi Gordji
AU - M. Filali
TI - Arens regularity of module actions
JO - Studia Mathematica
PY - 2007
VL - 181
IS - 3
SP - 237
EP - 254
AB - 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 $^{(n)}$ of . Banach algebras for which Z( **) = but $ ⊊ Z^{t}( **)$ are found (here Z( **) and $Z^{t}( **)$ 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 $Z(**) ≠ Z^{t}(**)$; (ii) $Z(**) = Z^{t}(**)$ 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.
LA - eng
KW - Arens product; topological centres; strongly Arens irregular; module actions; triangular Banach algebras
UR - http://eudml.org/doc/286315
ER -