Banach Algebra Research Articles

A system of additive functional equations in complex Banach algebras

Abstract In this article, we solve the system of additive functional equations: 2 f ( x + y ) − g ( x ) = g ( y ) , g ( x + y ) − 2 f ( y − x ) = 4 f ( x ) \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}2f\left(x+y)-g\left(x)=g(y),\\ g\left(x+y)-2f(y-x)=4f\left(x)\end{array}\right. and prove the Hyers-Ulam stability of the system of additive functional equations in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of f f -hom-ders in Banach algebras.

Spectral radius inequalities for matrices with entries from a Banach algebra

In this paper, we give spectral radius inequalities for matrices with entries from a Banach algebra. An emphasis will be given to matrices with commuting entries. Our inequalities are natural generalizations of earlier known inequalities for operator matrices.

$$L^\infty $$ estimates for the Banach-valued $${{\bar{\partial }}}$$-problem in a disk

We study the differential equation $$\frac{\partial G}{\partial {{\bar{z}}}}=g$$ with an unbounded Banach-valued Bochner measurable function g on the open unit disk $${\mathbb {D}}\subset {{\mathbb {C}}}$$ . We prove that under some conditions on the growth and essential support of g such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $${\mathbb {D}}$$ with values in a complex commutative unital Banach algebra.

Ideal Connes-Amenability of Certain Dual Banach Algebras

Ideal Connes-Amenability of Certain Dual Banach Algebras

Commutativity Theorems and Projection on the Center of a Banach Algebra

Abstract Let 𝑥 be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from 𝑥 to its center. On the other hand, we investigate the commutativity of 𝑥 under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.

Contributions to Automatic Continuity of $$(\sigma , \tau ) $$-Derivations on Banach Algebras

Contributions to Automatic Continuity of $$(\sigma , \tau ) $$-Derivations on Banach Algebras

Locally Attractivity Solution of Coupled Fractional Integral Equation in Banach Algebras

In this paper we prove the Locally Attractivity solution of coupled hybrid functional integral equation of fractional order (CQHFIE) in Banach algebras. Our main result is based on the hybrid fixed point theory used theorem due to Dhage B. C.

Fixed Point Theorems in Generalized Banach Algebras and an Application to Infinite Systems in 𝒞([0, 1], c 0) × 𝒞([0, 1], c 0)

The purpose of this paper is to extend Boyd and Wong’s fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra where c 0 is the space of all real sequences converging to zero.

On decomposition of left-right Atkinson linear relations and related classes

ABSTRACT In this paper, left Atkinson linear relations defined in a Banach space are considered. The behaviour of such notion in polynomials is studied and a spectral mapping theorem for the corresponding spectrum is deduced. Let denote any of the following classes: left (right) Atkinson, Fredholm, left (right) Weyl, Weyl, left (right) Browder, Browder linear relations. For a closed and everywhere defined linear relation we prove that under suitable conditions, where and is a bounded nilpotent operator if and only if admits a Kato-decomposition and 0 is not an interior point of the spectrum of As an application, some connections results between and are established, where denotes the Kato-decomposition spectrum of

Norm-controlled inversion in Banach algebras of integral operators

Norm-controlled inversion in Banach algebras of integral operators

On Harmonic Hilbert Spaces on Compact Abelian Groups

Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted L^2 spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach ^*-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group G to have the same spectrum as the C^*-algebra of continuous functions on G. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on L^2(G), and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

BSE-Properties of Vector-Valued Group Algebras

Let \(\mathcal {A}\) be a semisimple commutative Banach algebra with identity of norm one and G be a locally compact abelian group. In this paper, we study the BSE-Property for \(L^{1}(G,\mathcal {A})\) and show that \(L^{1}(G,\mathcal {A})\) is a BSE algebra if and only if \(\mathcal {A}\) is so.

Identities on additive mappings in semiprime rings

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

Invariant means on weakly almost periodic functionals with application to quantum groups

AbstractLet ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$ . For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$ , the space of weakly almost periodic functionals on ${\mathcal A}$ , admits a right (left) invariant $\varphi $ -mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$ -mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$ , for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$ . Finally, we study the existence of left and right invariant $1$ -means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ .

On Norm Estimates for Derivations in Norm-Attainable Classes

In this note, we provide detailed characterization of operators in terms of norm-attainability and norm estimates in Banach algebras. In particular, we establish the necessary and sufficient conditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.

Approximately invertible elements in non-unital normed algebras

We introduce the concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.

Action of projections on Banach algebras

<abstract><p>Let $ \mathcal{A} $ be a Banach algebra and $ n > 1 $, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such that $ \mathcal{P}(a^n) = a^n\; \text{for all} \; a \in \mathcal{G} $, the nonvoid open subset of $ \mathcal{A} $, then $ \mathcal{A} $ is commutative and $ \mathcal{P} $ is the identity mapping on $ \mathcal{A} $. Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research.</p></abstract>

New characterizations of g-Drazin inverse in a Banach algebra

In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a in a Banach algebra has g-Drazin inverse if and only if there exists x ? A such that ax = xa, a-a2x ? A qnil. As an application, we obtain the sufficient and necessary conditions for the existence of the g-Drazin inverse for certain 2 x 2 anti-triangular matrices over a Banach algebra. These extend the results of Koliha (Glasgow Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra, 27(1999), 3583-3592 and Zou et al. (Studia Scient. Math. Hungar., 54(2017), 489-508).

Characterisations of the radical in a Banach algebra

In this paper we characterise the radical in a Banach algebra A in terms of the perturbation ideal P(R) of a set R in A.

Induced topologies on certain Banach algebras

Let A be a Banach algebra with a bounded approximate identity bounded by 1. Two new topologies ?so and ?wo are introduced on A. We study these topologies and compare them with each other and with the norm topology. The properties of ?so and ?wo are then studied further and we pay attention to the group algebra L1(G) of a locally compact group G. Various necessary and sufficient conditions are found for a locally compact group G to be finite.