Biprojectivity and biflatness of amalgamated duplication of Banach algebras

2019 ◽  
Vol 19 (07) ◽  
pp. 2050132
Author(s):  
Ali Ebadian ◽  
Ali Jabbari
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Module Amenability ◽  
Amalgamated Duplication ◽  
Left And Right

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. We show that (super) amenability of [Formula: see text] implies (super) module amenability of [Formula: see text] and (super) amenability [Formula: see text]. We investigate biprojectivity and biflatness of [Formula: see text] in the some especial cases. We also give some results related to module biprojectivity and module biflatness of [Formula: see text], when [Formula: see text] is biprojective or biflat.

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.

The first module (σ,τ)-cohomology group of triangular Banach algebras of order three

2018 ◽  
Vol 17 (12) ◽  
pp. 1850225
Author(s):  
Hülya İnceboz ◽  
Berna Arslan
Keyword(s):  
Semigroup Forum ◽  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Module Structure ◽  
Semigroup Algebras ◽  
Module Amenability ◽  
Weak Module ◽  
Triangular Banach Algebras

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.

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}\} $.

scholarly journals Module amenability, module character biprojectivity and module character biflatness of lau product of two Banach algebras

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6627-6641
Author(s):  
H. Sadeghi ◽  
Bami Lashkarizadeh
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Module Amenability ◽  
Character Amenability ◽  
Module Homomorphism ◽  
Approximate Amenability

Let A be a Banach algebra and T be an U-module homomorphism from U-bimodule B into U-bimodule A. We investigate module amenability (resp. module approximate amenability), module character amenability (resp. module character approximate amenability), module character biprojectivity and module character biflatness of A x Tu B for every two Banach U-bimodule A and B.

scholarly journals The Structure ofφ-Module Amenable Banach Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mahmood Lashkarizadeh Bami ◽  
Mohammad Valaei ◽  
Massoud Amini
Keyword(s):  
Inverse Semigroup ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Module Amenability ◽  
Banach Modules ◽  
Finite Set

We study the concept ofφ-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions ofφ-amenability andφ-module amenability of Banach algebras. As a consequence, we show that, ifSis an inverse semigroup with finite setEof idempotents andl1Sis a commutative Banachl1E-module, thenl1S**isφ**-module amenable if and only ifSis finite, whenφ∈Homl1El1Sis an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).

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.

C∗-algebras defined by amalgamated duplication of C∗-algebras

2019 ◽  
pp. 2150019
Author(s):  
Ali Ebadian ◽  
Ali Jabbari
Keyword(s):  
Banach Algebra ◽  
Necessary Conditions ◽  
Approximate Identity ◽  
C Algebras ◽  
Sufficient And Necessary Conditions ◽  
Amalgamated Duplication ◽  
Left And Right

Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible action of [Formula: see text] on [Formula: see text]. We define [Formula: see text] as a [Formula: see text]-algebra, where it is a strongly splitting [Formula: see text]-algebra extension of [Formula: see text] by [Formula: see text]. Normal, self-adjoint, unitary, invertible and projection elements of [Formula: see text] are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of [Formula: see text] as a Banach algebra and as a [Formula: see text]-algebra are given. We characterize ∗-automorphisms on [Formula: see text] and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this [Formula: see text]-algebra. Also, we have constructed a new Hilbert [Formula: see text]-module [Formula: see text] over [Formula: see text], where [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text] and [Formula: see text] is a Hilbert [Formula: see text]-module over [Formula: see text].

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.

Weak module amenability of triangular banach algebras II

2019 ◽  
Vol 69 (2) ◽  
pp. 425-432
Author(s):  
Ebrahim Nasrabadi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Special Structure ◽  
Module Amenability ◽  
Weak Module ◽  
Triangular Banach Algebras

Abstract Let A and B be Banach 𝔄-bimodule and Banach 𝔅-bimodule algebras, respectively. Also let M be a Banach A, B-module and Banach 𝔄, 𝔅-module with compatible actions. In the case of 𝔄 = 𝔅, the author along with Pourabbas [5] have studied the weak 𝔄-module amenability of triangular Banach algebra $\begin{array}{} \displaystyle \mathcal{T}=\left[\begin{array}{rr} A & M \\ & B \end{array} \right] \end{array}$ and showed that 𝓣 is weakly 𝔄-module amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable, where A, B and M are unital. In this paper we investigate a special structure of 𝔄 ⊕ 𝔅-bimodule derivation from 𝓣 into 𝓣∗ and show that 𝓣 is weakly 𝔄 ⊕ 𝔅-bimodule amenable if and only if the corner Banach algebras A and B are weakly 𝔄-module amenable and weakly 𝔅-module amenable, respectively, where A, B and M are essential and not necessary unital.

scholarly journals Bounded Approximate Identities in Ternary Banach Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Madjid Eshaghi Gordji ◽  
Ali Jabbari ◽  
Gwang Hui Kim
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities ◽  
Left And Right

LetAbe a ternary Banach algebra. We prove that ifAhas a left-bounded approximating set, thenAhas a left-bounded approximate identity. Moreover, we show that ifAhas bounded left and right approximate identities, thenAhas a bounded approximate identity. Hence, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras.

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