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.

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 .

THE BOCHNER–SCHOENBERG-EBERLEIN PROPERTY OF EXTENSIONS OF BANACH ALGEBRAS AND BANACH MODULES

2021 ◽  
pp. 1-12
Author(s):  
NASRIN ALIZADEH ◽  
ALI EBADIAN ◽  
SAEID OSTADBASHI ◽  
ALI JABBARI
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Banach Modules ◽  
Module Extension

Abstract Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

scholarly journals On bounded dual-valued derivations on certain Banach algebras

2009 ◽  
Vol 86 (100) ◽  
pp. 107-114
Author(s):  
A.L. Barrenechea ◽  
C.C. Peña
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Banach Algebras ◽  
Class D ◽  
Unitary Dual ◽  
Dual Banach Algebra

We consider the class D(U) of bounded derivations Ud?U*defined on a Banach algebra U with values in its dual space U*so that ?x,d(x)? = 0 for all x?U U. The existence of such derivations is shown, but lacking the simplest structure of an inner one. We characterize the elements of D(U) if span(U2) is dense in U or if U is a unitary dual Banach algebra.

A Generalized Fredholm Theory for Certain Maps in the Regular Representations of an Algebra

1968 ◽  
Vol 20 ◽  
pp. 495-504 ◽  
Author(s):  
Bruce Alan Barnes
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Regular Representations ◽  
Completely Continuous Operators ◽  
Left And Right ◽  
Continuous Operators ◽  
Continuous Linear Operators

Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators.

scholarly journals ARENS REGULARITY AND AMENABILITY OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2012 ◽  
Vol 87 (2) ◽  
pp. 195-206 ◽  
Author(s):  
S. J. BHATT ◽  
P. A. DABHI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Arens Regularity ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractGiven a morphism T from a Banach algebra ℬ to a commutative Banach algebra 𝒜, a multiplication is defined on the Cartesian product space 𝒜×ℬ perturbing the coordinatewise product resulting in a new Banach algebra 𝒜×Tℬ. The Arens regularity as well as amenability (together with its various avatars) of 𝒜×Tℬ are shown to be stable with respect to T.

scholarly journals Biprojectivity and biflatness of generalized module extension Banach algebras

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5895-5905 ◽  
Author(s):  
Mina Ettefagh
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Module Extension

We investigate biprojectivity and biflatness of generalized module extension Banach algebra A Z B, in which A and B are Banach algebras and B is an algebraic Banach A-bimodule, with multiplication: (a, b)?(a',b') = (aa', ab' + ba' + bb')

A NOTE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2015 ◽  
Vol 92 (2) ◽  
pp. 282-289 ◽  
Author(s):  
F. ABTAHI ◽  
A. GHAFARPANAH
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Stability Result ◽  
Algebra Homomorphism ◽  
Arens Regularity ◽  
Algebra Morphism

Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

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