Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism

2015 ◽  
Vol 31 (9) ◽  
pp. 1461-1474 ◽  
Author(s):  
Prakash A. Dabhi ◽  
Ali Jabbari ◽  
Kazem Haghnejad Azar
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Weak Amenability ◽  
Algebra Morphism

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 .

Weak Amenability of a Class of Banach Algebras

2001 ◽  
Vol 44 (4) ◽  
pp. 504-508 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Dual Algebra ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that, if a Banach algebra is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of implies the (2m+ 1)-weak amenability of for all m ≥ 1.

scholarly journals Derivations into annihilators of the ideals of Banach algebras

2019 ◽  
Vol 52 (1) ◽  
pp. 20-28
Author(s):  
Akram Teymouri ◽  
Abasalt Bodaghi ◽  
Davood Ebrahimi Bagha
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Weak Amenability ◽  
New Concepts ◽  
Hereditary Properties ◽  
Ideal Amenability ◽  
The Ideal

AbstractIn this article, following Gorgi and Yazdanpanah, we define two new concepts of the ideal amenability for a Banach algebra A. We compare these notions with J-weak amenability and ideal amenability, where J is a closed two-sided ideal in A. We also study the hereditary properties of quotient ideal amenability for Banach algebras. Some examples show that the concepts of A/J-weak amenability and of J-weak amenability do not coincide for Banach algebras in general.

On amenability of the Banach algebras l∞(S, [Ufr ])

2002 ◽  
Vol 132 (2) ◽  
pp. 319-322
Author(s):  
FÉLIX CABELLO SÁNCHEZ ◽  
RICARDO GARCÍA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Short Note ◽  
Basic Information ◽  
Arens Product ◽  
Pointwise Multiplication ◽  
Weak Amenability ◽  
Usual Norm ◽  
Bounded Functions ◽  
Second Dual

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

Weak module amenability of triangular Banach algebras

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Abdolrasoul Pourabbas ◽  
Ebrahim Nasrabadi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Module Amenability ◽  
Weak Amenability ◽  
Weak Module ◽  
Triangular Banach Algebras

AbstractLet A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra $\mathcal{T} = \left[ {_B^{AM} } \right]$ and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When $\mathfrak{A}$ is a Banach algebra and A and B are Banach $\mathfrak{A}$-module with compatible actions, and M is a commutative left Banach $\mathfrak{A}$-A-module and right Banach $\mathfrak{A}$-B-module, we show that A and B are weakly $\mathfrak{A}$-module amenable if and only if triangular Banach algebra T is weakly $\mathfrak{T}$-module amenable, where $\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} $.

BIPROJECTIVITY AND BIFLATNESS OF LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

2014 ◽  
Vol 91 (1) ◽  
pp. 134-144 ◽  
Author(s):  
F. ABTAHI ◽  
A. GHAFARPANAH ◽  
A. REJALI
Keyword(s):  
Banach Algebra ◽  
Product Space ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Algebra Morphism ◽  
Cartesian Product Space

AbstractLet ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

scholarly journals A Generalization of Cyclic Amenability of Banach Algebras

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Behrouz Shojaee ◽  
Abasalt Bodaghi
Keyword(s):  
Banach Algebra ◽  
Homomorphic Image ◽  
Banach Algebras ◽  
Amenable Banach Algebra ◽  
Weak Amenability ◽  
Simon Stevin

AbstractThis paper continues the investigation of Esslamzadeh and the first author which was begun in [ESSLAMZADEH, G. H.-SHOJAEE, B.: Approximate weak amenability of Banach algebras, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), 415-429]. It is shown that homomorphic image of an approximately cyclic amenable Banach algebra is again approximately cyclic amenable. Equivalence of approximate cyclic amenability of a Banach algebra A and approximate cyclic amenability of M

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 More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

2018 ◽  
Vol 68 (1) ◽  
pp. 147-152
Author(s):  
Mohammad Ramezanpour
Keyword(s):  
Banach Algebra ◽  
Direct Product ◽  
Banach Algebras ◽  
Algebra Morphism

AbstractFor two Banach algebrasAandB, theT-Lau productA×TB, was recently introduced and studied for some bounded homomorphismT:B→Awith ∥T∥ ≤ 1. Here, we give general nessesary and sufficent conditions forA×TBto be (approximately) cyclic amenable. In particular, we extend some recent results on (approximate) cyclic amenability of direct productA⊕BandT-Lau productA×TBand answer a question on cyclic amenability ofA×TB.

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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