Let A be a factor von Neumann algebra with dimA ? 2. In this paper, it is proved that map : nonlinear mixed Jordan triple ?-derivation if and only an additive ?-derivation.

Let G = [ A M N B ] be a generalized matrix algebra defined by the Morita context (A,B,A MB,B NA,ΦMN ,ΨNM) . In this article we mainly study the question of whether there exist the so-called “proper” Jordan derivations for the generalized matrix algebra G . It is shown that if one of the bilinear pairings ΦMN and ΨNM is nondegenerate, then every antiderivation of G is zero. Furthermore, if the ...

Throughout this paper, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. As usual the commutator xy− yx will be denoted by [x, y]. We will use basic commutator identities [xy,z] = [x,z]y + x[y,z] and [x, yz] = [x, y]z+ y[x,z]. Recall that a ring R is prime if aRb = (0) implies that either a = 0 or b...

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...

We prove that every 2-local derivation from the algebra Mn(A)(n > 2) into its bimodule Mn(M) is a derivation, where A is a unital Banach algebra and M is a unital A-bimodule such that each Jordan derivation from A into M is an inner derivation, and that every 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie d...

We determine conditions under which a left Jordan derivation defined on an $MA$-semiring $S$ is this semiring and prove when implies the commutativity of $S$.

Let T be a triangular ring. An additive map δ from T into itself is said to be Jordan derivable at an element Z ∈ T if δ(A)B +Aδ(B) + δ(B)A+Bδ(A) = δ(AB+BA) for any A,B ∈ T with AB + BA = Z. An element Z ∈ T is called a Jordan all-derivable point of T if every additive map Jordan derivable at Z is a Jordan derivation. In this paper, we show that some idempotents in T are Jordan all-derivable po...

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free ∗-prime ring with involution is a generalized left derivation.

This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic structure of finite-dimensional quantum theory. The derivation relies on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit spaces having...