Let $R$ be a 2-torsion free ring and $U$ be a square closed Lie ideal of $R$. Suppose that $alpha, beta$ are automorphisms of $R$. An additive mapping $delta: R longrightarrow R$ is said to be a Jordan left $(alpha,beta)$-derivation of $R$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin R$. In this paper it is established that if $R$ admits an additive mapping $G : Rlongrigh...

let r be a prime ring with extended centroid c, h a generalized derivation of r and n ⩾ 1 a xed integer. in this paper we study the situations: (1) if (h(xy))n = (h(x))n(h(y))n for all x; y 2 r; (2) obtain some related result in case r is a noncommutative banach algebra and h is continuous or spectrally bounded.

Let A be a Banach algebra and X a Banach A-bimodule, the derivation D : A → X is semi-inner if there are ξ, μ ∈ X such that D(a) = a.ξ − μ.a, (a ∈ A). A is called semi-amenable if every derivation D : A → X∗ is semi-inner. The dual Banach algebra A is Connes semi-amenable (resp. approximately semi-amenable) if, every D ∈ Z1w _ (A,X), for each normal, dual Banach A-bimodule X, is semi -inner (re...

let $r$ be a 2-torsion free ring and $u$ be a square closed lie ideal of $r$. suppose that $alpha, beta$ are automorphisms of $r$. an additive mapping $delta: r longrightarrow r$ is said to be a jordan left $(alpha,beta)$-derivation of $r$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin r$. in this paper it is established that if $r$ admits an additive mapping $g : rlongrigh...

A Banach algebra A is weakly amenable provided that every bounded derivation from A to its dual A is inner. In H1], the rst-named author, building on earlier work of J. W. Bunce and W. L. Paschke BP], proved that every C-algebra is weakly amenable. We give a simpliied and uniied proof of this theorem. B. E. Johnson has proved that every bounded Jordan derivation from a C-algebra A to any Banach...

Let A be a C-algebra acting on a Hilbert space H, σ : A → B(H) be a linear mapping and d : A → B(H) be a σ-derivation. Generalizing the celebrated theorem of Sakai, we prove that if σ is a continuous ∗-mapping then d is automatically continuous. In addition, we show the converse is true in the sense that if d is a continuous ∗-σ-derivation then there exists a continuous linear mapping Σ : A → B...

Let A be a C-algebra acting on a Hilbert space H, σ : A → B(H) be a linear mapping and d : A → B(H) be a σ-derivation. Generalizing the celebrated theorem of Sakai, we prove that if σ is a continuous ∗-mapping then d is automatically continuous. In addition, we show the converse is true in the sense that if d is a continuous ∗-σ-derivation then there exists a continuous linear mapping Σ : A → B...

using fixed pointmethods, we investigate approximately higher ternary jordan derivations in banach ternaty algebras via the cauchy functional equation$$f(lambda_{1}x+lambda_{2}y+lambda_3z)=lambda_1f(x)+lambda_2f(y)+lambda_3f(z)~.$$

Recently, Masanobu Kaneko introduced a conjecture on an extension of the derivation relations for multiple zeta values. The aim of this paper is to give a proof of the conjecture by reducing it to a class of relations for multiple zeta values studied by Kawashima. Also we will give some algebraic aspects of the extended derivation operator ∂ (c) n on Q〈x, y〉, which was defined by modeling a Hop...