Arens regularity and local reflexivity principle for Banach algebras

1989 ◽  
Vol 284 (1) ◽  
pp. 23-40 ◽  
Author(s):  
B. Iochum ◽  
G. Loupias
Keyword(s):  
Banach Algebras ◽  
Arens Regularity ◽  
Local Reflexivity

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

2018 ◽  
Vol 17 (09) ◽  
pp. 1850169 ◽  
Author(s):  
Hossein Javanshiri ◽  
Mehdi Nemati
Keyword(s):  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Multiplier Algebra ◽  
Topological Center ◽  
Maximal Ideals ◽  
Open Questions ◽  
Arens Regularity ◽  
Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

scholarly journals φ-Multipliers on Banach Algebras and Topological Modules

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Marjan Adib
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Topological Module ◽  
Topological Modules ◽  
Arens Regularity

We prove some results concerning Arens regularity and amenability of the Banach algebraMφAof allφ-multipliers on a given Banach algebraA. We also considerφ-multipliers in the general topological module setting and investigate some of their properties. We discuss theφ-strict andφ-uniform topologies onMφA. A characterization ofφ-multipliers onL1G-moduleLpG, whereGis a compact group, is given.

scholarly journals Amenability and topological centres of the second duals of Banach algebras

2002 ◽  
Vol 65 (2) ◽  
pp. 191-197 ◽  
Author(s):  
F. Ghahramani ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dual Algebra ◽  
Arens Product ◽  
Weak Amenability ◽  
Arens Regularity ◽  
Second Dual

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

scholarly journals ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

2008 ◽  
Vol 50 (3) ◽  
pp. 539-555 ◽  
Author(s):  
MATTHEW DAWS
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Main Tool ◽  
Bilinear Map ◽  
General Will ◽  
Arens Product ◽  
C Algebras ◽  
Local Reflexivity

AbstractThe Arens products are the standard way of extending the product from a Banach algebrato its bidual″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that ifis Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated With Locally Compact Groups

2021 ◽  
Author(s):  
Anthony To-Ming Lau ◽  
Ali ÜLger
Keyword(s):  
Banach Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Bounded Operators ◽  
Unified Approach ◽  
Locally Compact ◽  
Topological Center ◽  
Power Bounded Operators ◽  
Arens Regularity ◽  
Power Bounded

Abstract Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.

Arens regularity of module actions and weak amenability of Banach algebras

2015 ◽  
Vol 71 (2) ◽  
pp. 224-235 ◽  
Author(s):  
Ali Bagheri Vakilabad ◽  
Kazem Haghnejad Azar ◽  
Ali Jabbari
Keyword(s):  
Banach Algebras ◽  
Weak Amenability ◽  
Arens Regularity

scholarly journals Arens Regularity of Certain Class of Banach Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Abbas Sahleh ◽  
Abbas Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Banach Algebras ◽  
Arens Regularity ◽  
Left And Right ◽  
Module Extension

We study Arens regularity of the left and right module actions of on , where is thenth dual space of a Banach algebra , and then investigate (quotient) Arens regularity of as a module extension of Banach algebras.

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