On the distance of the composition of two derivations to the generalized derivations

Abstract

Introduction.A well-known theorem of E. Posner [10] to the set of all generalized derivations of A. We consider the case when A is an ultraprime normed algebra (Theorem 1), the case when d x = d 2 and A is ultrasemiprime (Theorem 2), and finally, the case when A is a von Neumann algebra (Theorem 3). As a consequence of these results we obtain a partial answer to Mathieu's question [8]: what is the norm of the composition of two derivations in a prime C*-algebra?Our results will follow easily from two entirely elementary observations; it is our aim in this paper to point out the kind of method that could be used.

Derivation rings of Lie rings

“…An additive mapping F : A → A is called a generalized derivation of A associated with a derivation δ ∈ Der A (in the sence of Brešar [4]) if…”

Commutativity of $\ast$-Prime Rings with Generalized Derivations

“…This observation leads to the following definition, given in [5]; an additive mapping F : R À3 R is called a generalized derivation with associated derivation d if F(xy) F(x)y xd(y) holds for all x; y P R:…”

SOME COMMUTATIVITY THEOREMS OF PRIME RINGS WITH GENERALIZED (Σ, Τ)-Derivation

“…Recently, in [7], Bresar defined the following notation. An additive mapping f : R → R is called a generalized derivation if there exists a derivation d : R → R such that f (xy) = f (x)y + xd(y) for all x, y ∈ R.…”