Banach Algebra Research Articles

The order bidual of C(X) for a realcompact space

It is well known that the bidual of C(X) for a compact space X, supplied with the Arens product, is isometrically isomorphic as a Banach algebra to for some compact space . The space is unique up to homeomorphism. We establish a similar result for realcompact spaces: The order bidual of C(X) for a realcompact space X, when supplied with the Arens product, is isomorphic as an f-algebra to for some realcompact space . The space is unique up to homeomorphism.

Generalized domination of ergodic elements in ordered Banach algebras

Let A be an ordered Banach algebra and a, b ∈ A such that 0 ≤ a ≤ b. The domination problem involves finding natural conditions under which properties of b will be inherited by a. In [12] this problem was investigated for ergodic elements, where a ∈ A is ergodic if the sequence converges in A. In this paper we consider the more general problem where the assumption 0 ≤ a ≤ b is replaced by the weaker condition ±a ≤ b.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Character Contractibility and Amenability of Banach Algebras with Applications to Quantum Groups

Character Contractibility and Amenability of Banach Algebras with Applications to Quantum Groups

Integral of Ordered Banach Algebra Valued Measurable Functions

This research is concerned with the set of functions in ordered Banach algebra values that links up between functional analysis and measure theory. We generalized the concept of integration by using the measure space and the measurable function where is an ordered Banach algebra by using the integration of a simple measurable function with values ​​in an ordered Banach algebra space (represented by an indicator function that has values ​​in an ordered Banach algebra) and the integral of a non-negative measurable function that has values ​​in an ordered Banach algebra. The aim of this research is to define the integration of functions by using the measure in the ordered Banach algebra space. This study generalized the definition of integration for the measurable function with values in the ordered Banach algebra space.

A fixed point technique to the stability of Hadamard 𝔇-hom-der in Banach algebras

Abstract In this paper, we introduce the concept of Hadamard 𝔇-hom-der on Banach algebras. Further, we apply the fixed point method to prove the stability of the Hadamard 𝔇-hom-der on Banach algebras.

A Pexider system of additive functional equations in Banach algebras

AbstractIn this paper, we solve the system of functional equations $$\begin{aligned} \textstyle\begin{cases} f(x+y)+g(y-x)=2f(x), \\ g(x+y)-f(y-x)=2g(y) \end{cases}\displaystyle \end{aligned}$$ { f ( x + y ) + g ( y − x ) = 2 f ( x ) , g ( x + y ) − f ( y − x ) = 2 g ( y ) and we investigate the stability of g-derivations in Banach algebras.

Schweizer-Sklar power aggregation operators based on complex intuitionistic fuzzy information and their application in decision-making

In 1960, Schweizer and Sklar introduced the novel Schweizer-Sklar t-norm and t-conorm which is used in the construction of aggregation operators. Schweizer-Sklar norms are more general than algebraic norms and Einstein norms. Additionally, computing the power operators based on the Schweizer-Sklar norms for complex Atanassov intuitionistic fuzzy (CA-IF) set is very awkward and complicated. In this manuscript, firstly, we propose the Schweizer-Sklar operational laws for CA-IF values, and secondly, we develop the CA-IF Schweizer-Sklar power averaging (CA-IFSSPA) operator, CA-IF Schweizer-Sklar power ordered averaging (CA-IFSSPOA) operator, CA-IF Schweizer-Sklar power geometric (CA-IFSSPG) operator, and CA-IF Schweizer-Sklar power ordered geometric (CA-IFSSPOG) operator. Some suitable and dominant properties for the above operators are also discussed. Furthermore, to simplify the above operators, we develop the procedure of decision-making technique, called multi-attribute decision-making (MADM) methods based on the proposed operators based on CA-IF values. Finally, we compare the proposed methods with some existing methods to describe the efficiency and capability of the discovered approaches by some examples.

Some results on ultradifferential convolution property of ultradistributions

We consider some sequential conditions for the convolvability of two ultradistributions, specifically the Roumieu ultradistributions defined on the space of all ultradifferentiable functions Dn{M_p} in the space Dn'{M_p} and show the similarities between these conditions in the sense of Mincheva-Kaminska S. The sequential conditions are based on the use of approximate identities which satisfy certain conditions to be a Banach algebra for the space Dn{M_p} . We present an equivalent result involving the convolution of the differentiability of two Roumieu ultradistributions with equivalent norm condition. 

Permuting triderivations and permuting trihomomorphisms in complex Banach algebras

Permuting triderivations and permuting trihomomorphisms in complex Banach algebras

Zero product determined of amalgamated duplication of Banach algebras

In this paper, we study some sufficient and necessary conditions for the amalgamated duplication Banach algebra [Formula: see text] to be zero product determined. Precisely, we show that in the case where [Formula: see text] has a bounded approximate identity, [Formula: see text] is zero product determined if and only if [Formula: see text] is zero product determined and [Formula: see text] has the property [Formula: see text]. Moreover, we give a better characterization when [Formula: see text] is unital. As an application, we also construct some examples on semidirect product of Banach algebras and the [Formula: see text]–Lau product of Banach algebras. Finally, we raise some open problems related to the zero product determined of classical module extension Banach algebras.

Free Probability on Banach Algebras Induced by Scaled Hypercomplex Numbers

Free Probability on Banach Algebras Induced by Scaled Hypercomplex Numbers

The stability of Jordan k-*-derivations on Γ∗-Banach algebras by fixed point method

Using fixed point methods, we prove the stability and the superstability of Jordan k-∗-derivations on Γ∗-Banach algebras for the following Jensen-type functional equation μf((x+y)/2)+μf((x-y)/2)=f(μx) where μ is a complex number such that |μ| = 1. We also investigate the stability and the superstability of Jordan k-∗-derivations with the functional equation f(2μx+μy)+f(μx+2μy)=μ[f(3x)+f(3y)] on Γ∗-Banach algebras.

Numerical representation of topological real algebras

We show that the isomorphism provided by the Gelfand-Mazur theorem for a commutative pre-ordered real Banach algebra A with unit defines a numerical representation which is compatible with the order structure.

Spectrum, homomorphisms and multipliers of Lau product of Banach algebras

Given Banach algebras $A$, $B$ and a continuous homomorphism $\theta:B\longrightarrow A$ with $\Vert\theta\Vert \leq1$, we obtain characterization of spectrum, homomorphisms and multipliers of $A\times_{\theta}B$, which is a strongly splitting Banach algebra extension of $B$ by $A$. Also we characterize the semisimplicity of these algebras.

Hyper-instability of Banach algebras

<abstract><p>In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.</p></abstract>

On the sheafyness property of spectra of Banach rings

AbstractLet be a non‐Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to a homotopical Huber spectrum via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces that is a homeomorphism in some well‐known examples of non‐sheafy Banach rings, where is the usual Huber spectrum of . This permits the use of the tools from derived geometry to understand the geometry of in cases when the classical structure sheaf is not a sheaf.

On the stability of cubic bi-derivations on Banach algebras

On the stability of cubic bi-derivations on Banach algebras

Fixed point theorems for $ b $-generalized contractive mappings with weak continuity conditions

<abstract><p>The purpose of this paper was to introduce several $ b $-generalized contractive mappings in the framework of cone $ b $-metric spaces over Banach algebras. The obtained contractions generalized and extended the counterparts in metric spaces, cone metric spaces, and $ b $-metric spaces. Moreover, via weakening the completeness of the spaces, we gave some fixed point theorems for asymptotically regular mappings without considering the orbital continuity and $ k $-continuity of the mappings. Those who need a specification is our results do not rely on the continuity of $ b $-metric and the normality of cones. In addition, some nontrivial examples were presented to illustrate the superiority of our fixed point theorems.</p></abstract>

An extension of Herstein's theorem on Banach algebra

<abstract><p>Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.</p></abstract>