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.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

scholarly journals The Technique of Measures of Noncompactness in Banach Algebras and Its Applications to Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Józef Banaś ◽  
Szymon Dudek
Keyword(s):  
Banach Algebra ◽  
Integral Equations ◽  
Banach Algebras ◽  
Functional Integral ◽  
Existence Results ◽  
Measures Of Noncompactness ◽  
Nonlinear Functional ◽  
Half Axis ◽  
Functional Integral Equations

We study the solvability of some nonlinear functional integral equations in the Banach algebra of real functions defined, continuous, and bounded on the real half axis. We apply the technique of measures of noncompactness in order to obtain existence results for equations in question. Additionally, that technique allows us to obtain some characterization of considered integral equations. An example illustrating the obtained results is also included.

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 CHARACTERIZATION OF SOME BIDERIVATIONS ON TRIANGULAR BANACH ALGEBRAS

2021 ◽  
pp. 929
Author(s):  
Sedigheh Barootkoob
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Triangular Banach Algebras ◽  
Apparent Similarity

Let $A$ and $B$ be unital Banach algebras‎, ‎$X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A‎, ‎B$ and $X$‎. The structure of some derivations on triangular Banach algebras was studied by some authors. ‏‎Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations‎, ‎there are fundamental differences between them‎. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we ‎completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$‎ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$‎‏.

scholarly journals Estimations des solutions de l'équation de Bezout dans les algèbres de Beurling analytiques

2005 ◽  
Vol 96 (2) ◽  
pp. 307 ◽  
Author(s):  
O. El-Fallah ◽  
M. Zarrabi
Keyword(s):  
Banach Algebra ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Bezout Equation ◽  
Beurling Algebras

Let $A$ be a unitary commutative Banach algebra with unit $e$. For $f\in A$ we denote by $\hat f$ the Gelfand transform of $f$ defined on $\hat A$, the set of maximal ideals of $A$. Let $(f_1,\dots,f_n)\in A^n$ be such that $\sum_{i=1}^n\|f_i\|^2 \leq 1$. We study here the existence of solutions $(g_1,\dots,g_n)\in A^n$ to the Bezout equation $f_1g_1+\cdots+f_ng_n=e$, whose norm is controlled by a function of $n$ and $\delta=\inf_{\chi\in\hat A}(|\hat f_1(\chi)|^2+\cdots+|\hat f_n(\chi)|^2)^{1/2}$. We treat this problem for the analytic Beurling algebras and their quotient by closed ideals. The general Banach algebras with compact Gelfand transform are also considered.

scholarly journals Characterization of ideals of rhotrices over a ring and its applications

2021 ◽  
Vol 27 (1) ◽  
pp. 138-147
Author(s):  
Kailash M. Patil ◽  
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Higher Order ◽  
Unital Ring ◽  
Maximal Ideals ◽  
Unital Banach Algebra

We define higher order rhotrices over a commutative unital ring S and obtain a ring \mathcal{R}_n(S) of rhotrices of the order n \in \mathbb{N}. We characterize the ideals and maximal ideals of \mathcal{R}_n(S). As a particular case, we record ideals of rhotrix rings over integers and rhotrix algebras over complex plane \mathbb{C}. As an application, we characterize the maximal ideals of the commutative unital Banach algebra \mathcal{R}_n(\mathbb{C}).

scholarly journals A Characterization of Bi-Jordan Homomorphisms on Banach Algebras

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Abbas Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms

For Banach algebras A and B, we show that if U=A×B is unital and commutative, each bi-Jordan homomorphism from U into a semisimple commutative Banach algebra D is a bihomomorphism.

Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations

2020 ◽  
Vol 70 (4) ◽  
pp. 1003-1011
Author(s):  
Behrooz Fadaee ◽  
Kamal Fallahi ◽  
Hoger Ghahramani
Keyword(s):  
Banach Algebra ◽  
Complex Variable ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Jordan Derivation ◽  
Continuous Linear ◽  
Linear Map ◽  
Standard Operator ◽  
Standard Operator Algebras

AbstractLet 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

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.

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