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In this work, we define the set which consists of n-normal operators. Also, the sets and and defined from. Firstly, the set is proved Banach algebra and establish consists of n-normal operators. Secondly is shown C* -algebra, also some properties and theories are proved to reach a circular for the spectral theory. Finally, this study is established and proved generalization the spectral theory of n-normal operators. [ABSTRACT FROM AUTHOR]

In this work, we define the set which consists of (n, m)-normal operators. Also, the sets and and defined from. Firstly, the set is proved Banach algebra and establish consists of (n, m)-normal operators. Secondly, we give theorem of the Gelfand representation of the Banach algebra generated by a (n, m)-normal operator, also some theories are proven to reach a circular for the spectral theory. Finally, this study is established and proved generalization the spectral theory of (n, m)-normal operators. [ABSTRACT FROM AUTHOR]

Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved "abstract" Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries. [ABSTRACT FROM AUTHOR]

Let ℜ be a ring with center Z(ℜ). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 such that ([𝔉(sm)sn + sn𝔉(sm), sr]k)l ∈ Z(ℜ) or every s ∈ ℜ, then either 𝔉(s) = ps for every s ∈ ℜ or ℜ satisfies s4 and 𝔉(s) = sp for every s ∈ ℜ and p ∈ 𝔘, the Utumi quotient ring of ℜ. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map. [ABSTRACT FROM AUTHOR]

We introduce and study a new class of generalized inverses in rings. An element a in a ring R has generalized Zhou inverse if there exists b ∈ R such that bab = b , b ∈ com m 2 (a) , a n − a b ∈ J (R) for some n ∈ N. We prove that a ∈ R has generalized Zhou inverse if and only if there exists p = p 2 ∈ com m 2 (a) such that a n − p ∈ J (R) for some n ∈ N. Cline's formula and Jacobson's lemma for generalized Zhou inverses are established. In particular, the Zhou inverse in a ring is characterized. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of algebras satisfying certain conditions which has been named power algebras and we prove some fixed point theorems for a new class of contractions. We give new conditions for the existence and uniqueness of fixed points for this new class of mappings which are not a contraction in the usual algebra. Also, we introduce a new iterative algorithm for fixed-point problems in a Banach algebra. This results improve and extend the corresponding conclusions of the Ishikawa algorithm under weaker conditions and lead to stronger results. In conclusion, we give an application to nonlinear integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate three new versions of the Rickart condition for rings. These conditions, as well as, three new corresponding regularities are defined using projection invariance. We show how these conditions relate to each other as well as their connections to the well-known Baer, Rickart, quasi-Baer, p.q.-Baer, regular, and biregular conditions. Applications to polynomial extensions and to triangular and full matrix rings are provided. Examples illustrate and delimit results. [ABSTRACT FROM AUTHOR]

In this article we discuss the commutativity of a prime Banach algebra A with the help of its generalized (α , α) -derivation. In particular, we prove that if A is a unital prime Banach algebra and A has a nonzero continuous linear generalized (α , α) -derivation g associated with a nonzero continuous linear (α , α) -derivation d such that g ((x y) n) - d (x n) d (y n) = 0 or g ((x y) n) - d (y n) d (x n) = 0 for sufficiently many x, y and integer n = n (x , y) > 1 then A must be commutative. In this connection some more results has been found. Further examples are given to show that hypotheses are not superfluous. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra over R or C. In this paper, we describe the behavior of recently defined b -generalized skew derivations which satisfy certain differential identities on some specific subsets of A. [ABSTRACT FROM AUTHOR]

Let A be a Banach space and λ be a non-zero fixed element of A ∗ (dual space of A) with non-zero kernel. Defining algebra product in A as a · b = λ(a)b for a, b ∈ A, we show that A is a (2m− 1)-weakly amenable Banach algebra but not 2m-weakly amenable for any m ∈ N. Furthermore, we show the converse of the statement [2, Proposition 1.4.(ii)] “for a non-unital Banach algebra A, if A is weakly amenable then A# is weakly amenable” does not hold. [ABSTRACT FROM AUTHOR]

Consider the space L of entire functions represented by multiple Dirichlet series that becomes a non uniformly convex Banach space which is also proved to be dense, countable and separable. Continuing further, for the given space L the characterization of bounded linear transformations in terms of matrix and characterization of linear functional has been obtained. [ABSTRACT FROM AUTHOR]

This article covers studying the description of 2 local bilateral symmetric multiplications in Banach algebra of functional component matrix. The definitions related to this are mentioned, and the lemma and theorem are proven. [ABSTRACT FROM AUTHOR]

In this paper, we provide examples which show that there exists a type of approximate identity between the class of bounded weak approximate identities and bounded approximate identities. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concepts of Jordan amenability and Jordan biflatness of Banach algebras and compare those with the classical notions of amenability and biflatness. We also find some relations between Jordan amenability and the existence of Jordan approximate and virtual diagonals. These could be considered as Jordan versions of the classical results due to Johnson and Helemskii. We show that, for all C*-algebras, the concepts of amenability and Jordan amenability coincide. [ABSTRACT FROM AUTHOR]

For n ≥ 2 , an additive map f between two rings A and B is called an n-Jordan homomorphism, or an n-homomorphism if f (a n) = f (a) n , for all a ∈ A , or f (a 1 a 2 ⋯ a n) = f (a 1) f (a 2) ⋯ f (a n) , for all a 1 , a 2 , ... , a n ∈ A , respectively. In particular, if n = 2 then f is simply called a Jordan homomorphism or a homomorphism, respectively. The notion of n-Jordan homomorphism between rings was introduced in 1956 by Herstein and the concept of n-homomorphism between algebras was introduced in 2005 by Hejazian et al. Properties of n-Jordan homomorphisms as well as n-homomorphisms have been studied by many authors since then. One of the main questions is that, "under what conditions n- Jordan homomorphisms are n-homomorphism?". Another natural question is that "under what conditions certain properties of homomorphisms may be extended to n-homomorphisms". We provide conditions under which these questions have affirmative answers. We also study the continuity problem for n-Jordan homomorphisms on Banach algebras, while extending some known results in this field. [ABSTRACT FROM AUTHOR]

The generalized notion of weak amenability, namely $(\varphi,\psi)$-weak amenability, where $\varphi,\psi$ are continuous homomorphisms on a Banach algebra ${\mathcal A}$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi,\psi)$-weak amenability on the measure algebra $M(G)$, the group algebra $L^1(G)$ and the Segal algebra $S^1(G)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi,\psi)$-weak amenability of a special semigroup algebra is shown as well.

We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to \(L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)\), where \(\rho:\mathbb{R}_+\to\mathbb{R}_+\) is a monotonically nondecreasing function from some family \(\mathscr{R}\). In particular, \(\mathscr{R}\) includes the functions \(\rho(x)=(1+x)^{\alpha}\) with \(\alpha\geq 1\).

In the present paper, our main work aims to discover the existence result of the fractional order non-linear Hadamard functional integral equations on [1,a] by employing the theory of measure of non-compactness together with the fixed point theory in Banach space. An example is presented to see the validity and practicability of our existence result.

Abstract While the Bedrosian identity for the Hilbert transform of a product does not hold for general Sobolev class functions, we show that the defect of this identity is more regular than would be naively expected. We use this result to give a stronger-than-expected estimate on the chain rule defect of the square root of the Laplacian.

The aim of this paper is to prove the superstability of the following functional equations: f (P (x , y)) = g (x) h (y) , f (x + y) = g (x) h (y). [ABSTRACT FROM AUTHOR]

Let S be a foundation topological semigroup and M a (S) the space of all measures μ ∈ M (S) for which the maps x ⟼ | μ | ∗ δ x and x ⟼ δ x ∗ | μ | from S into M(S) are weakly continuous. In the present paper, we introduce and study the concept of ϕ -amenability for S and investigate the relations between ϕ -amenability of S and essential ϕ ^ -amenability of M a (S) , where ϕ is a character on S and ϕ ^ is the extension of ϕ to M a (S) . [ABSTRACT FROM AUTHOR]

In this paper, we will deal with perturbation theory in the Banach algebra of 2 × 2 matrices with entries in a Banach algebra A. Also we study some essential spectra of a 2 × 2 matrix operators with respect to a subalgebra. [ABSTRACT FROM AUTHOR]

The relation , where is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that and are elements of a Banach algebra (or of an Arens–Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of degenerates, but at each zero of , some local algebra of power series remains. Moreover, this local algebra depends only on the order of the zero. As an application, we prove a result about closed subalgebras of holomorphically finitely generated algebras. [ABSTRACT FROM AUTHOR]

Segal extensions, not necessarily injective, of a Banach algebra into a Banach algebra are discussed resulting into answering when do the Gelfand map of a commutative Banach algebra as well as Gelfand-Naimark construction on a, not necessarily commutative, Banach ∗ -algebra give Segal extensions. Analogous to C ∗ -Segal algebras arising from Segal extension into C ∗ - algebras, uB-Segal algebras arising from Segal extensions into uniform Banach algebras are intrinsically characterized. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions for hybrid fractional differential equations involving Caputo fractional derivative of order 2 < ζ 3. We base our arguments on a generalization of Darbo’s fixed point theorem combined with the approaches related with measures of noncompactness in Banach algebras. To demonstrate the argument, an illustration is provided. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider the concept of cone b-metric spaces over Banach algebras and obtain some fixed point results for various definitions of contractive mappings. Moreover, we discuss about the property P and the property Q of fixed point problems. Our results are significant, since we omit the assumptions of normality of cones under which can be generalized and unified a number of recently announced results in the existing literature. In particular, we refer to the results of Huang et al. [H. Huang, G. Deng, S. Radenovic, Some topological properties and fixed point results in cone metric spaces over Banach algebras, Positivity. (2018), in press]. [ABSTRACT FROM AUTHOR]

Let (X, d) be a compact metric space. Then, the normed algebra of pointwise Lipschitz functions on X is denoted by Lip pw (X , d). We study and characterize the completeness of these algebras and obtain a necessary and sufficient condition for such an algebra to be complete. We then investigate the endomorphisms of Lip pw (X , d) by considering their associated self-maps. A necessary and sufficient condition is given for these operators to be compact. Moreover, we discuss relations between Lip pw (X , d) and other normed function algebras. [ABSTRACT FROM AUTHOR]

This paper addresses the characterization of the multipliers for Gelfand pairs. More precisely we obtain a theorem of Wendel type for the space of integrable bi-invariant functions on a locally compact group. We also obtain some results concerning the double multipliers for Gelfand pairs. [ABSTRACT FROM AUTHOR]

In this paper, with the help of a variant of Schauder fixed point theorem in the real Banach algebra together with the finite difference method (FDM), we take a brief look at the p -adic analog of Richards' equation derived by Khrennikov et al. [Application of p -adic wavelets to model reaction–diffusion dynamics in random porous media, J. Fourier Anal. Appl.22 (2016) 809–822], and study the solvability and solution of this problem. This equation is formulated by a kinetic equation during the modeling of the reaction–diffusion dynamics in random porous media. Moreover, in order to guarantee the convergence of the presented iterative schemes, some sufficient conditions would be presented. [ABSTRACT FROM AUTHOR]

In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1

Suppose that $A$ is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra $C_{rm{BSE}}(Delta(A))$ consisting of all BSE-functions on $Delta(A)$ where $Delta(A)$ denotes the character space of $A$. Indeed, in the case that $A=C_0(X)$ where $X$ is a non-empty locally compact Hausdroff space, we give a complete characterization of $Delta(C_{rm{BSE}}(Delta(A)))$ and in the general case we give a partial answer. Also, using the Fourier algebra, we show that $C_{rm{BSE}}(Delta(A))$ is not a $C^*$-algebra in general. Finally for some subsets $E$ of $A^*$, we define the subspace of BSE-like functions on $Delta(A)cup E$ and give a nice application of this space related to Goldstine's theorem.

Let A be a Banach algebra, X be a Banach left A-module and n ≥ 2 be an integer. A bounded linear operator T: A → X is called an n-Jordan multiplier if for each a ∈ A, T(an)=a· T(an-1). In this paper we investigate this notion and give some illuminating examples. Also, we give an approximate local version of n-Jordan multipliers and try to investigate when an approximate local n-Jordan multiplier is an n-Jordan multiplier. Finally, for functional Banach algebras we give a characterization of n-Jordan multipliers.

Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k {\rm e}^{\langle\lambda^k, z\rangle}$ for which $(|\lambda^k|/{\rm e})^{|\lambda^k|} k!|a_k|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

The inverse scattering problem for half-line Schrödinger operators with potentials from the Marchenko class is shown to be closely related to some Banach algebra of functions on the line. In particular, it is proved that the topological conditions in the Marchenko theorem can be replaced by the condition that the scattering function should belong to this Banach algebra.

The main purpose of this paper is to present some fixed point results concerning the generalized Reich type α-admissible mappings in cone b-metric spaces over Banach algebras. Our results are significant extensions and generalizations of resent results of N. Hussain at al. (2017) and many well-known results in abundant literature. We also gave an example that confirmed our results.

The purpose of the manuscript is to establish the solvability of generalized nonlinear functional integral equations including the kernel Ψ-functions in Banach algebra. We prove that such a proposed functional integral equation has at least one solution in the space C(I, R), where I = [0, 1]. The measure of non-compactness in C(I, R) together with Darbo's fixed point approach are applied to prove the results. Finally, an example is given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We give examples showing that the usual Artin approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular, we construct an example of a commutative integral Banach algebra R such that the ring of formal power series over R is not flat over the ring of convergent power series over R. [ABSTRACT FROM AUTHOR]

In the present paper we obtain a condition on vector valued coefficients of multiple Dirichlet series for when the series converges in the whole complex plane. We also prove some results related to Banach algebraic structure, topological divisor of zero and more on a class of such series satisfying certain condition. [ABSTRACT FROM AUTHOR]

Let 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established. [ABSTRACT FROM AUTHOR]

In this paper, we characterize closed ideals with bounded approximate identities in some Banach algebras. Our results provide answers to a question of Lau and Ülger. [ABSTRACT FROM AUTHOR]

An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions F(x,y) defined on the extended real plane [-∞,∞]² that vanish when x or y is -∞. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative ∂2/(∂x∂y) of this space of primitives. If f=∂2/(∂x∂y)F then the integral over interval [a,b]×[c,d]⊆[-∞,∞]² is ∫ba∫dcf=F(a,c)+F(b,d)-F(a,d)-F(b,c) and ∫∞-∞∫∞-∞f=F(∞,∞). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is ∥f∥=∥F∥∞ where F is the unique primitive of f. The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both L1 and the space of Henstock--Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in ∥⋅∥∞ are also inherited by the integrable distributions. It is shown that the dual space are the functions of bounded Hardy--Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder inequality, second mean value theorem, Fubini theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in Rn are sketched out. [ABSTRACT FROM AUTHOR]