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Let A be a Banach algebra and X be a locally compact space. We derive various character amenability and construct the corresponding approximate character means of the A-valued continuous function algebra C0(X,A) directly from those of A. [ABSTRACT FROM AUTHOR]

In this paper, we present a general version of the algebra AM(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ℳ(A), the multiplier algebra of A, and obtain Banach algebra AM. Then, we study the spaceability of AM− A and AM (G) ℒ ℒA(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that AM(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of AM(G). [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.

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.

Introduction and preliminaries Hilbert C∗-modules were firrst introduced in the work of I. Kaplansky.Hilbert C*-modules are the natural generalization that of Hilbert spaces arising by replacing of the field of scalars C by a C∗-algebra. Let us recall some basic facts about the Hilbert C∗-modules. Let A be a C∗-algebra. An right inner product A-module is a linear space X which is a right A-module (with compatible scalar multiplication: λ(x.a) =(λx).a = x.(λa) for x ∈ X, a ∈ A, λ ∈ C), together with a map (x, y)→ : X × X → A such that for all x, y, z ∈ X, a ∈ A, α, β ∈ C./files/site1/files/72/10Abstract.pdf

For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

Let α be a fixed complex number, and let Ω be a simply connected region in complex plane C that is starlike with respect to α ∈ Ω. We define some Banach space of analytic functions on Ω and prove that it is a Banach algebra with respect to the α-Duhamel product defined by ... . We prove that its maximal ideal space consists of the homomorphism hα defined by hα(f)=f(α). Further, we characterize the lattice of invariant subspaces of the integration operator Jαf(z)=... f(t)dt. Moreover, we describe in terms of α-Duhamel operators the extended eigenvectors of Jα. [ABSTRACT FROM AUTHOR]

An asymptotic expansion of the solution of a nonhomogeneous matrix difference equation of general form is obtained. The case when there is no bound on the differences of the arguments is considered. The effect of the roots of the characteristic equation is taken into account. The asymptotic behavior of the remainder is established depending on the asymptotics of the free term of the equation. [ABSTRACT FROM AUTHOR]

In a Banach algebra A it is well known that the usual spectrum has the following property: σ (a b) \ { 0 } = σ (b a) \ { 0 } for elements a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. X ⊂ A such that for a , b ∈ A : 1 - a b ∈ X ⇒ 1 - b a ∈ X. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence results for a class of hybrid initial value problems of Riemann-Liouville fractional differential equations. Our investigation is based on the Dhage hybrid fixed point theorem, remarks and some special cases will be discussed. The continuous dependence of the unique solution on one of its functions will be proved.

Abstract 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 ) . $$\begin{aligned}& f \bigl(P(x,y) \bigr)= g(x)h(y), \\& f(x+y)=g(x)h(y). \end{aligned}$$

For p ∈ [ 1 , ∞) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2. Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2. In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K -theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2. Our methods also allow us to recover a folklore fact in the case of C*-algebras (p = 2), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras. [ABSTRACT FROM AUTHOR]

Introduction The notion of weak amenability for commutative Banach algebras was introduced and studied for the first time by Bade, Curtis and Dales. Johnson extended this concept to the non commutative case and showed that group algebras of all locally compact groups are weakly amenable. A Banach algebra A is called weakly amenable if every continuous derivation D:A→A* is inner. It is often useful to restrict one's attention to derivations D:A→A* satisfying the property Dac+Dca=0 for all a,c∈A. Such derivations are called cyclic. Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable if every continuous cyclic derivations D:A→A*is inner. This notion was presented by Gronbaek. He investigated the hereditary properties of this concept, found some relations between cyclic amenability of a Banach algebra and the trace extension property of its ideals. Ghahramani and Loy introduced several approximate notions of amenability by requiring that all bounded derivations from a given Banach algebra A into certain Banach A-bimodules to be approximately inner. In the same paper and the subsequent one, the authors showed the distinction between each of these concepts and the corresponding classical notions and investigated properties of algebras in each of these new classes. Motivated by this notions, Esslamzadeh and Shojaee defined the concept of approximate cyclic amenability for Banach algebras and investigated the hereditary properties for this new notion. Periliminaries Let A be a Banach algebra and let X be an A-bimodule. Then the l1-direct sum A×X under the multiplication a,xb,y=ab,ay+xb (a,b∈A,x,y∈X), is a Banach algebra called the module extension of A by X and denoted by A⊕X. The class of module extension Banach algebras contains a wide variety of Banach algebras includes a triangular Banach algebra Tri(A,X,B). Every triangular Banach algebra Tri(A,X,B).can be identified with the module extension Banach algebra (A×B)⊕X. On the other hand, for two Banach algebra A and B with ∆(B)≠∅ and for θ∈∆(B), the set of all non-zero multiplicative linear functionals on B, the θ-Lau product A×θB is a Banach algebra which is defined as the l1-direct sum A×B equipped with the algebra multiplication a1,b1a2,b2=a1a2+θb2a1+θb1a2,b1b2 a1,a2∈A,b1,b2∈B. This type of product was introduced by Lau for certain class of Banach algebras known as Lau algebras and was extended by Sangani Monfared for arbitrary Banach algebras. The unitization A♯ of A can be regarded as the ι-Lau product A×ιC, where ι∈Δ(C) is the identity map. This product provides not only new examples of Banach algebras by themselves, but it can also serve as a source of (counter) examples for various purposes in functional and harmonic analysis. From the homological algebra point of view A×θB is a strongly splitting Banach algebra extension of B by A. The Lau product of Banach algebras enjoys some properties that are not shared in general by arbitrary strongly splitting extensions. For instance, commutativity is not preserved by a generally strongly splitting extension. However, A×θB is commutative if and only if both A and B are commutative. Results and discussion Many basic properties of A♯, some notions of amenability and some homological properties are extended to A×θB by many authors. In particular, Ghaderi, Nasr-Isfahani and Nemati extended some results on (approximate) cyclic amenability of A♯, obtained by Esslamzadeh and Shojaee, to A×θB. They showed that if A2 is dense in A then the cyclic amenability A×θB is equivalent to the cyclic amenability of both A and B. In this paper, by characterizing of cyclic derivations on Lau product A×θB and module extension A⊕X, we present general necessary and sufficient conditions for those to be (approximate) cyclic amenable. This not only provides new results on (approximate) cyclic amenability of these type of Banach algebras but also improves some main results in this topic. In particular we show that, under mild condition, the cyclic amenability of Tri(A,X,B) is equivalent to the cyclic amenability of the corner algebras A and B. [ABSTRACT FROM AUTHOR]

t is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natural multiplications extending the one on A. But the essence of A* is completely different. By defining new products, we investigate some algebraic and spectral properties of odd duals of A. We will show relations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A*, and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras. [ABSTRACT FROM AUTHOR]

We investigate under what conditions n-Jordan homomorphisms between rings are n-homomorphism, or homomorphism; and under what conditions, n-Jordan homomorphisms are continuous. One of the main goals in this work is to show that every n-Jordan homomorphism f : A → B, from a unital ring A into a ring B with characteristic greater than n, is a multiple of a Jordan homomorphism and hence, it is an n-homomorphism if every Jordan homomorphism from A into B is a homomorphism. In particular, if B is an integral domain whose characteristic is greater than n, then every n-Jordan homomorphism f : A → B is an n-homomorphism. Along with some other results, we show that if A and B are unital rings such that the characteristic of B is greater than n, then every unital n-Jordan homomorphism f : A → B is a Jordan homomorphism and hence, it is an m-Jordan homomorphism for any positive integer m ≥ 2. We also investigate the automatic continuity of n-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the n-Jordan homomorphism has dense range. [ABSTRACT FROM AUTHOR]

Using generalized Caputo fractional left and right vectorial Taylor formulae, we establish mixed fractional Ostrowski and Grüss type inequalities involving several Banach algebra valued functions. The estimates are with respect to all norms ∥·∥p, 1 ≤p ≤∞.

For a Banach algebra A and a norm one character ϕ on A , we characterize ϕ -amenability and ϕ -contractibility of A in terms of metrically homological properties of some Banach A -modules. Also, for any Banach A -module, metrically injectivity and projectivity of submodules and quotient modules are investigated. Finally, we give some results on metrically injectivity of A ∗ and its submodules. [ABSTRACT FROM AUTHOR]

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of infinite matrices, sequences and infinite series are in K. In this paper, following ([4], [5]), we prove some more Steinhaus type theorems over K. [ABSTRACT FROM AUTHOR]

This note aims to present some rudimentary aspects of relatively bounded nonlinear (not necessarily linear) operators. To be precise, we prove that the set of all nonlinear operators between two normed linear spaces that are relatively bounded with respect to a fixed operator between these spaces itself is a normed linear space under a suitable norm. Furthermore, it is shown that if the codomain is a Banach space, then the space of all relatively bounded nonlinear operators is also complete. Some elementary results on the algebraic structure of the space of all nonlinear operators that are bounded with respect to a fixed operator are also recorded. The results reported in this note illustrate that with some predictable minor modifications, the fundamental classical results on the space of all bounded linear operators can be carried over to a more general setting. Here the word 'general' is used in a double sense that the boundedness is generalized to relative boundedness and the restriction of linearity is dropped to include nonlinear maps. [ABSTRACT FROM AUTHOR]

kkllk|Using a generalized vectorial Taylor formula involving ordinary vector derivatives we establish mixed Ostrowski, Opial and Hilbert-Pachpatte type inequalities for several Banach algebra valued functions. The esti- mates are with respect to all norms ||·||p, 1 ≤ p ≤ ∞. We finish with applications. [ABSTRACT FROM AUTHOR]

Using generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert-Pachpatte type inequalities for Banach algebra valued functions. We cover also the sequential fractional case. We finish with applications. [ABSTRACT FROM AUTHOR]

Let A be a complex Banach algebra. An element a ∈ A has g-Drazin inverse if there exists b ∈ A such that b = b a b , a b = b a , a − a 2 b ∈ A q n i l. New additive results for the g-Drazin inverse in a Banach algebra are presented. These extend the main results of Deng and Wei (J. Math. Analysis Appl., 379(2010), 313–321) and Wang et al. (Filomat, 30(2016), 1185–1193). [ABSTRACT FROM AUTHOR]

In this manuscript, by removing the domain convexity hypothesis, the existence of fixed set results for the sum and the product of (p + 1)-multi-valued operators ... A · Bi, acting on Banach algebras satisfying a sequential condition (P) under weak topology is proved. In addition, by using a new definition of the multi-valued operator (I/A), we obtain new fixed-set theorems for the operators of the form (I/A)- ... Bi under some suitable conditions on the operators A, B1, ... Bp. Applications to self-similarity theory are also given. [ABSTRACT FROM AUTHOR]

Employing generalized Canavati fractional left and right vectorial Taylor formulae we prove corresponding left and right fractional Hilbert-Pachpatte type inequalities for von Neumann-Schatten class Bγ (H) valued functions. We cover also the sequential fractional case. We finish with applications. [ABSTRACT FROM AUTHOR]

We study the algebra of vector valued multipliers of Banach algebra valued McShane integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, then functions associated with measures of strong bounded variation and the set {L∞([α, b], R) e} are vector valued multipliers of McShane integrable functions. We find some necessary and another set of sufficient conditions for a function g to define a multiplier. In case X satisfies Radon Nikodym property (weak Radon Nikodym property), we study multiplier operators. [ABSTRACT FROM AUTHOR]

Al-Banach algebra was introduced in 1940 by Russian mathematician I.M. Gelfand and arose from the observation that some Banach spaces exhibit interesting properties when they can be provided with operations and examples. Among the linear functions adopted in Al-Banach commutative algebra and its maxima, as well as with the spectra of its elements in one way or another that leads us towards the central component of the spectra mapping theory and the spectral radius formula. In this paper, we get some results on the spectral radius of Al-Banach algebra. [ABSTRACT FROM AUTHOR]

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Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M ∈ A n × n , M ˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M ∈ C n × n , σ (M) ⊂ C denotes the set of eigenvalues of M. It is shown that if A ∈ A n × n and B ∈ A m × m are such that for all φ ∈ M A , σ (A ˆ (φ)) ∩ σ (B ˆ (φ)) = ∅ , then for all C ∈ A n × m , the Sylvester equation A X − X B = C has a unique solution X ∈ A n × m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra. [ABSTRACT FROM AUTHOR]

Let A and U be Banach algebras such that U is also a Banach A -bimodule with compatible algebra operations, module actions and compatible norm. By defining an appropriate multiplication, we turn ℓ 1 -direct product A × U into a Banach algebra so that A is a closed subalgebra and U is a closed ideal of it. This algebra is, in fact, the semidirect product of A and U which we denote by A ⋉ U . In this paper, we study automatic continuity of derivations on A ⋉ U in a general setting. As an application of our results, we present various results about the automatic continuity of derivations of module extension Banach algebras and θ -Lau products of Banach algebras. Some examples are also given. [ABSTRACT FROM AUTHOR]

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions. [ABSTRACT FROM AUTHOR]

We introduce the e–core–EP inverse and the core–EP inverse for elements of Banach algebras. Several characterizations and representations for these inverses are presented using idempotents and Drazin inverses. Then, we define the e–core inverse of elements in Banach algebras and give its characterizations, generalizing the related results of the core inverse. [ABSTRACT FROM AUTHOR]

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D: A ↦ (A/JA)∗ if D∗∗(A∗∗) ⊆ WAP (A/JA), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A** is weak module amenable then A is also weak module amenable. [ABSTRACT FROM AUTHOR]

Using the techniques of Darbo's fixed-point theorem associated with measures of noncompactness, we investigate the existence results for functional stochastic integral equations in Banach Algebra, which are useful in many real-life problems, such as applied mathematics, physics, and economics problems. Our existence results treat numerous existence results achieved by different authors as special cases. An example is also provided to show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

It is shown that the Banach algebra C ub (X , d) of bounded uniformly K -valued continuous functions on a metric space (X, d) is coherent if and only if d is a uniformly discrete metric, or equivalently, if X does not contain twin sequences. The proof is based on Neville’s result that C (X , R) for a Tychonov space is coherent if and only if X is basically disconnected. Since C ub (X , d) is self-adjoint, we also include in the survey part some general results on the spectrum M(A) of general self-adjoint Banach function algebras which we need for the special case here. [ABSTRACT FROM AUTHOR]

Introduction Let , and be Banach spaces and be a bilinear mapping. In 1951 Arens found two extension for as and from into . The mapping is the unique extension of such that from into is continuous for every , but the mapping is not in general continuous from into unless . Thus for all the mapping is continuous if and only if is Arens regular. Regarding as a Banach , the operation extends to and defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space becomes a Banach algebra. Material and methods The constructions of the two Arens multiplications in lead us to definition of topological centers for with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive, -algebra is Arens regular. In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras. Results and discussion We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity. Conclusion The following conclusions were drawn from this research. if and only if the bilinear mapping is Arens regular. A bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact. if and only if the bilinear mapping is Arens regular. Assume that has approximate identity. Then is Arens regular if and only if is reflexive../files/site1/files/62/9Abstract.pdf

Let $\left\{a_\alpha\right\}_{\alpha\in I}$ be a bounded net in a Banach algebra $A$ and $\varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $\|aa_\alpha-\varphi(a)a_\alpha\|\to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such as Segal algebras and $L^1$-algebras are responsive to this concept. It is also shown that $Wap(A)$ has a left invariant $\varphi$-mean if and only if there exists a bounded net $\left\{a_\alpha\right\}_{\alpha\in I}$ in $\left\{a\in A;\ \varphi(a)=1\right\}$ such that $\|aa_\alpha-\varphi(a)a_\alpha\|_{Wap(A)}\to0$ uniformly for all $a$ in weakly compact subsets of $A$. Other results in this direction are also obtained.

In this paper, we discuss some properties of joint spectral {radius(jsr)} and generalized spectral radius(gsr) for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but some are different. For example for a bounded set of scalar matrices,$\Sigma$, $r_*\left(\Sigma\right)= \hat{r}\left(\Sigma\right)$, but for a bounded set of upper triangular matrices with entries in a Banach algebra($\Sigma$), $r_*\left(\Sigma\right)\neq\hat{r}\left(\Sigma\right)$. We investigate when the set is defective or not and equivalent properties for having a norm equal to jsr, too.

Abstract By a stochastic controller, we make stable the pseudo stochastic Lie bracket (derivation, derivation) in complex MB-algebras. Next, we get an approximation by a stochastic Lie bracket (derivation, derivation) and calculate the maximum error of the estimate.

In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.

We take into consideration generalized derivations. First, we study the stability of generalized derivations on Banach algebras under consideration. Then we prove some theorems involving approximate generalized derivations on Banach algebras. These results can be applied to C*-algebras.