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In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. [ABSTRACT FROM AUTHOR]

In this paper, we study a connection between the operator Riccati equation S′(x) = KS(x) + S(x)K - 2S(x)KS(x), x ∈ R, and the set of reflectionless Schr¨odinger operators with operator-valued potentials. Here K ∈ B(H), K > 0 and S: R → B(H), where B(H) is the Banach algebra of all linear continuous operators acting in a separable Hilbert space H. Let S+(K) be the set of all solutions S of the Riccati equation satisfying the conditions 0 < S(0) < I and S′(0) ≥ 0, with I being the identity operator in H. We show that every solution S ∈ S+(K) generates a reflectionless Schr¨odinger operator with some potential q that is an analytic function in the strip... [ABSTRACT FROM AUTHOR]

In this paper, using fixed point method, we investigate the stability and also the superstability of cubic bi-derivations on Banach algebras. [ABSTRACT FROM AUTHOR]

Let 푅 be a prime ring and 퐿 a non-central Lie ideal of 푅. In this paper, we aim to classify the generalized derivations of 푅 satisfying some algebraic identities with power values on 퐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra. [ABSTRACT FROM AUTHOR]

Let R be a prime ring and L a Lie ideal of R. The purpose of this paper is to classify generalized derivations F 1 , F 2 , F 3 of R satisfying the following differential identity F 1 (x) ⊥ F 2 (y) = F 3 (x) ⊥ y f o r a l l x , y ∈ L , where ⊥ represents either the Lie product [.,.], or the Jordan product ∘ . Furthermore, as an application, the same identities are studied locally on nonvoid open subsets of prime Banach algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of Connes biprojectivity for a dual Banach algebra A with respect to its w∗-closed ideal I, say I-Connes biprojectivity. Some Lipschitz algebras Lipα(X) and some matrix algebras are studied under this new notion. Also, with some mild assumptions, the relation between IConnes biprojectivity and left ϕ-contractibility is given, where ϕ is a ω∗-continuous multiplicative linear functional on A. As an application, we characterize Connes biprojectivity of some Lipschitz algebras. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra. An element a ϵ A has generalized Drazin inverse if there exists b ϵ A such that b = bab, ab = ba, a - a²b ϵ Aqnil. New additive results for the generalized Drazin inverse of an operator over a Banach space are presented. we extend the main results of a paper of Shakoor, Yang and Ali from 2013 and of Wang, Huang and Chen from 2017. Appling these results to 2 x 2 operator matrices we also generalize results of a paper of Deng, Cvetković-Ilić and Wei from 2010. [ABSTRACT FROM AUTHOR]

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In 1989, Bakhtin proved a version of the Banach contraction principle in the context of b-metric spaces. More precisely, let (X, d; s) be a complete b-metric space with parameter s ⩾ 1 and let T a contractive map on X, that is a self-map T of X satisfying d(Tx, Ty) ⩽ λd(x, y), ∀x, y ∈ X, (Bλ) with some λ ∈ [0, 1). Bakhtin proved that if λ ∈ [0, 1 s), then T has a unique fixed point. In the two last decades many research papers were produced by many authors in the setting of b-metric spaces. In 2021, the author published a note on the Banach contraction principle in b-metric spaces in Divilgaciones Mat'ematicas dealing with some complements to Bakhtin results. The aim of this paper is to provide other complements. In particular, we discuss the remaining case where λ ∈ [1 s, 1). We give an evaluation of the order of convergence for the iterative Picard process and a posteriori error estimate for this process and we estimate the diameter of the T-orbits. We investigate well-posedness of the fixed problem of a map T satisfying Bλ when X is Torbitally complete and we establish two shadowing properties for these maps without requiring completeness of X. [ABSTRACT FROM AUTHOR]

In this paper, we find solutions and investigate the superstability bounded by a function (Gǎvruta sense) for the p -power-radical functional equation related to sine function equation: f ( x p + y p 2 p) 2 − f ( x p − y p 2 p) 2 = f (x) f (y) from an approximation of the p -power-radical functional equation: f ( x p + y p 2 p) 2 − f ( x p − y p 2 p) 2 = g (x) h (y) , where p is a positive odd integer, and f , g and h are complex valued functions on R . Furthermore, the obtained results are extended to Banach algebras. In this paper, we find solutions and investigate the superstability bounded by a function (Gǎvruta sense) for the p -power-radical functional equation related to sine function equation: f ( x p + y p 2 p) 2 − f ( x p − y p 2 p) 2 = f (x) f (y) from an approximation of the p -power-radical functional equation: f ( x p + y p 2 p) 2 − f ( x p − y p 2 p) 2 = g (x) h (y) , where p is a positive odd integer, and f , g and h are complex valued functions on R . Furthermore, the obtained results are extended to Banach algebras. [ABSTRACT FROM AUTHOR]

This paper investigates the commutativity of Banach algebras in which the group of invertible elements is subject to certain algebraic and metric conditions. Application to a particular class of Banach algebras is examined. [ABSTRACT FROM AUTHOR]

In this paper we make an attempt to prove common fixed point theorem for two and three mappings in Banach algebra by using measure of non-compactness. In the second section of the paper we prove the common fixed point theorem for two continuous linear operators on Banach algebra and also an effort is made to prove common fixed point theorem for three continuous commuting operators in Banach algebra. [ABSTRACT FROM AUTHOR]

Let A be a complex unital Banach algebra and D be a non-empty open connected subset of C. Let f be a holomorphic A-valued function on D. We set: ∑(f) = {z ∈ D | f(z) ∉ ∈ Inv(A)}, where Inv(A) is the set of all invertible elements in A. In this paper, we give a description of ∑(f) using the classical spectrum of f(z) (z ∈ D). Moreover, within the framework of this paper, we give a partial positive answer to the following problem which was posed by B. Aupetit: If the usual spectrum σ(f(z)) is polar for all z ∈ D, is it true that ∑(f) is polar? [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer and A be a C ∗ -algebra. A permuting n-linear map G : A n → A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D : A n → A such that G ς 1 , ς 2 , ... , ς i ς i ′ , ... , ς n = G ς 1 , ς 2 , ... , ς i , ... , ς n ς i ′ + ς i D (ς 1 , ς 2 , ... , ς i ′ , ... , ς n) holds ∀ ς i , ς i ′ ∈ A . In this paper, we investigate the structure of C ∗ -algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set, A be a commutative Banach algebra, and 1 ≤ p < ∞. In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of ℓ p X , A . Specifically, we establish that ℓ p X , A possesses the BSE property if and only if X is finite and A is BSE. Additionally, we investigate the BSE module property on Banach ℓ p X , A -modules and demonstrate that a Banach space ℓ p X , Y serves as a BSE Banach ℓ p X , A -module if and only if X is finite and Y represents a BSE Banach A -module. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the existence and uniqueness of continuous solutions of the following iterative functional differential equation ... By using of Boyd-Wong's fixed point theorem and under suitable conditions, we establish the existence and uniqueness of a continuous solution. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to generalize the concept of classical fuzzy set to vector-valued fuzzy set which can attend values not only in the real interval [ 0 ,   1 ] , but in an ordered interval of a Banach algebra as well. This notion allows us to introduce the concept of vector-valued fuzzy metric space which generalizes, extends and unifies the notion of classical fuzzy metric space and complex-valued fuzzy metric space and permits us to consider the fuzzy sets and metrics in a larger domain. Some topological properties of such spaces are discussed and some fixed point results in this new setting are proved. Multifarious examples are presented which clarify and justify our claims and results. [ABSTRACT FROM AUTHOR]

In this paper, we show that some additive mappings of a Banach algebra are derivations. Moreover, we exhibit some commutativity criteria for a prime Banach algebra admitting a derivation satisfying certain algebraic identities. [ABSTRACT FROM AUTHOR]

In this paper, we study and compare the types of diagonals for Lau product Banach algebras × φ ℬ and diagonals of two Banach algebras and ℬ , where φ is a homomorphism from ℬ to . As a consequence, we obtain the (approximate) amenability and contractibility × φ ℬ in terms of (approximate) amenability and contractibility of and ℬ. Moreover, we introduce the notion of pseudo-virtual diagonal for a Banach algebra and present some characterization results for Lau product Banach algebras. [ABSTRACT FROM AUTHOR]

Let 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 A is a prime Banach algebra admitting a continuous projection P with image in Z (A) such that P (a n) = a n for all a ∈ G , the nonvoid open subset of A , then A is commutative and P is the identity mapping on 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. [ABSTRACT FROM AUTHOR]

Let A be an algebra with a right identity. In this paper, we study (p, q)−centralizers of A and show that every (p, q)−centralizer of A is a two-sided centralizer. In the case where, A is normed algebra, we also prove that (p, q)−centralizers of A are bounded. Then, we apply the results for some group algebras and verify that L 1 (G) has a nonzero weakly compact (p, q)−centralizer if and only if G is compact and the center of L 1 (G) is non-zero. Finally, we investigate (p, q)−Jordan centralizers of A and determine them. [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results. [ABSTRACT FROM AUTHOR]

The Bochner–Schoenberg–Eberlein (BSE)-property on commutative Banach algebras is a property related to multiplier algebras of Banach algebras. In this paper, we answer the problem (12) raised by Javanshiri and Nemati in [Amalgamated duplication of the Banach algebra along a -bimodule , J. Algebra Appl. 17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the amalgamated Banach algebra ⋊ is BSE-algebra if and only if and are BSE-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we solve the system of functional equations { 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. [ABSTRACT FROM AUTHOR]

In this paper, the existence of the solutions of a class of weakly singular integral equations in Banach algebra is investigated. The basic tool used in investigations is the technique of the measure of non-compactness and Petryshyn's fixed point theorem. Also, for the applicability of the obtained results, some examples are given. [ABSTRACT FROM AUTHOR]

Introduction The notion of amenability for Banach algebras was first introduced and initiated by Johnson [3]. Afterwards, Gronbaek [2] studied the concept of cyclic amenability and it‘s hereditary properties of Banach algebras. Many Authors study and investigate the several notions of σ-amenability, for more details refer to [1], [5], [6], [8] and [9]. In the throughout of this paper we assume that A is a Banach algebra, I is a closed two-sided ideal in A, σ is a continuous homomorphism on A and X is an A-bimodule Banach. The linear map D: A→X is called derivation if D(ab) = D(a)∙b + a∙D(b) for all a,bϵA. Moreover, the linear mapping D: A → X is called a σ-derivation if D(ab) = D(a)∙σ(b) + σ(a)∙D(b) for each a,bϵA. A linear map D : A → X is called σ-inner derivation if there exists x1∈X such that D = adσx1, where adσx1(a) = σ(a).x1-x1.σ(a), for all a∈A. The Banach algebra A is called σ-amenable if every σ-derivation D : A → X∗ is σ-inner. This notion was first introduced and studied by Moslehian and Motlagh [9]. Recall that the derivation D:A→A∗ is cyclic if +=0, (a,bϵA). A is cyclic amenable if every cyclic derivation D:A→A∗ is inner. So, by using these notion we introduced the new notions σ-cyclic derivation and σ-cyclic amenability for Banach algebras. We say that the σ-derivation D : A → A∗ is σ-cyclic if ⟨D(a), σ(b)⟩ + ⟨D(b), σ(a)⟩ = 0, (a,bϵA). Also, the Banach algebra A is σ-cyclic amenable if every σ-cyclic derivation D:A→A∗ is σ-inner. The closed two-sided ideal I in A has the trace extension property if for each λϵI∗ with a∙λ=λ∙a (aϵA), there exists ΛϵA∗ such that Λ|I = λ and Λ∙a=a∙Λ, for all aϵA. Every homomorphism σ on A can be extended to a homomorphism σ̂: A/I → A/I suth that for aϵA defined by σ̂(a+I)=σ(a)+I. Next, we present the results obtained in this paper. We showed that let A/I be σ̂-cyclic amenable. Then I has the trace extension property. Also, Suppose that σ has dense range on A, I has the trace extension property and Ais σ-cyclic. Then A I is σ̂-cyclic amenable. Recall that the Banch algebra A is essential, if A̅̅2 = A. We showed that every σ-cyclic amenable Banach algebra is essential. If σ(I) ⊆ I, we can restrict σ to I and denoted it by σI :I → I. It is clear that σI is continuous homomorphism on I. Thus, if σ(I) ⊆ I, I is σI -cyclic amenable and A/I is σ̂-cyclic amenable, then A is σ-cyclic amenable. For the unitization of Banach algebra A i.e. A#=A ⊕ ℂe, and for every continuous homomorphism σ on A there exists a continuous homomorphism σ# : A# → A# such that defined by σ# (a, λe)=σ(a)+λe, (aϵA, λϵℂ). Therefore, with above notes, A is σ-cyclic amenable if and only if A# be σ# -cyclic amenable. Finally, for two Banach algebras A and B and nonzero multiplicative linear functional θ on B, the Cartesian space A×B with norm and multiplication ||(a,b)||=||a||+||b||, (a,b)(a’,b’)=(aa’+θ(b)a’+θ(b’)a, bb’), is a Banach algebra and denoted by A×𝜃𝐵. If σ and τ are homomorphisms on A and B respectively, then (σ, τ) denoted by <(σ, τ),(a, b)>=(σ(a), τ(b)) is a homomorphism on A×𝜃𝐵 if and only if θ∘τ=θ. By using this we showed that A×𝜃𝐵 is (σ, τ)-cyclic amenable if and only if A is σ-cyclic amenable and B is τ-cyclic amenable. [ABSTRACT FROM AUTHOR]

In this paper we try to approximate any properties of quasi-block Toeplitz matrix (QBT), by means of a finite number of parameters. A quasi-block Toeplitz (QBT) matrix is a semi-infinite block matrix of the kind F = T(F) + E where... These matrices are a Banach algebra with the standard arithmetic operations. We try to analysis some structures and computational properties for arithmetic operations of QBT matrices with some MATLAB commands. [ABSTRACT FROM AUTHOR]

In this paper, we introduce triple θ-centralizers and weak triple θ-centralizers on an algebra A, where θ: A → A is a triple homomorphism. Some observations concerning triple θ-centralizers, weak triple θ-centralizers and approximate weak triple θ-centralizers are given. [ABSTRACT FROM AUTHOR]

The equivariant coarse Baum--Connes conjecture interpolates between the Baum--Connes conjecture for a discrete group and the coarse Baum--Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum--Connes conjecture holds for (X; Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to find a representation of the Fock space on C n in terms of the weighted Bergman spaces of the projective spaces CP n − 1 ; i.e., every function in the Fock space can be written as a direct sum of elements in weighted Bergman space on CP n − 1 . Also, we study the C * algebras generated by Toeplitz operators where the symbols are taken from the following two families of functions: Firstly, the symbols depend on the moment map associated with the unit circle, and secondly, the symbols are invariant under the same action. Moreover, we analyze the commutative relations between these algebras, and we apply these results to find new commutative Banach algebras generated by Toeplitz operators on Fock space of C n . [ABSTRACT FROM AUTHOR]

The focus of this paper is to extend some stability results of the analysis of perturbed operators on the Banach space to the case of Fredholm perturbation theory in a Banach algebra relative to a bounded homomorphism. Our approach is to define some subsets in a Banach algebra to enrich the theory of perturbed left-right Browder and left-right Weyl elements in a Banach algebra relative to a bounded homomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the commutativity of a Banach algebra provided with a continuous derivation satisfying algebraic identities involving nonvoid open subsets of . Furthermore, we provide examples to show that various restrictions in the hypothesis of our theorems are not superfluous. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the general solution and the Hyers-Ulam stability of the system of biadditive functional equations ... in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of f -biderivations in complex Banach algebras. [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]

Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some user-friendly versions of integral-type fixed-point results and give some modifications of the classical Banach contraction principle by constructing a special type of contractive restrictions of integral forms for weak contraction mappings defined on 1 E -complete multiplicative partial cone metric spaces over Banach algebras and formulate some existence and uniqueness results regarding the fixed-point theorems using some integrative conditions. Moreover, we validate the significance our results and exploit them to find the unique solution of a fractional nonlinear differential equation of Caputo type, which complements some previously well-known generalizations found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new variants of fixed point theorems for a large class of countably nonexpansive multi-valued mappings. Some fixed point theorems for the sum and the product of three multi-valued mappings defined on nonempty, closed convex set of Banach algebras are also presented. These results improve and complement a number of earlier works. As an application, we prove existence results for a broad class of nonlinear functional integral inclusions as well as nonlinear differential inclusions. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach Algebras. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions. Moreover, it derives sufficient conditions for solution uniqueness and establishes the Hyers–Ulam types of solution stability. This study contributes valuable insights into complex hybrid fractional order systems and their practical implications. [ABSTRACT FROM AUTHOR]

In this paper, we consider rings of multisets consisting of elements of a Banach algebra. We investigate the algebraic and topological structures of such rings and the properties of their homomorphisms. The rings of multisets arise as natural domains of supersymmetric functions. We introduce a complete metrizable topology on a given ring of multisets and extend some known results about structures of the rings to the general case. In addition, we consider supersymmetric polynomials and other supersymmetric functions related to these rings. This paper contains a number of examples and some discussions. [ABSTRACT FROM AUTHOR]

In this paper, I introduce the properties EPCℵ1$\mathrm{EPC}_{\aleph _1}$ and ProjCes(E)$\mathrm{ProjCes}(E)$ for forcing notions and show that it is consistent that the forcing axiom for EPCℵ1+ProjCes(E)$\mathrm{EPC}_{\aleph _1}+ \mathrm{ProjCes}(E)$ forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property β1$\beta _1$. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex‐valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set E$E$ is an example of an EPCℵ1+ProjCes(ω1∖E)$\mathrm{EPC}_{\aleph _1}+ \mathrm{ProjCes}(\omega _1 \setminus E)$ forcing notion. As a corollary, it is consistent that a nonfree Whitehead group exists, the continuum hypothesis fails, and an ultrapower of the field of reals has the property β1$\beta _1$. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized Fourier-Feynman transform (GFFT) for functions on the general Wiener space Ca,b[0,T]. We establish an explicit evaluation formula for the analytic GFFT of bounded cylinder functions on Ca,b[0, T]. We start by examining certain cylinder functions which belong in a Banach algebra of bounded functions on Ca,b[0,T]. We then obtain an explicit formula for the analytic GFFT of the bounded cylinder functions. [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors (A,B)\mapsto A\otimes _{\alpha } B, where A\otimes _\alpha B is a cross norm completion of A\odot B for each pair of C*-algebras A and B. For the first class of bifunctors considered (A,B)\mapsto A{\otimes _p} B (1\leq p\leq \infty), A{\otimes _p} B is a Banach algebra cross-norm completion of A\odot B constructed in a fashion similar to p-pseudofunctions \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p-pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider {\otimes _{p,q}} for Hölder conjugate p,q\in [1,\infty ] – a Banach *-algebra analogue of the tensor product {\otimes _{p,q}}. By taking enveloping C*-algebras of A{\otimes _{p,q}} B, we arrive at a third bifunctor (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra. For G_1 and G_2 belonging to a large class of discrete groups, we show that the tensor products \mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2) coincide with a Brown-Guentner type C*-completion of \mathrm \ell ^1(G_1\times G_2) and conclude that if 2\leq p'

In this paper, we introduce the notion of graphical cone metric spaces over Banach algebra and prove some fixed point results for a particular type of contractive mappings defined on such spaces. These results extend and generalize several results from metric, graphical metric, and cone metric spaces. Some examples that demonstrate the results proved herein are provided. An application of our results to the existence of solution of a pair of initial value problems is provided. [ABSTRACT FROM AUTHOR]