Commutators with finite spectrum
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.
Commuting maps on lie ideals
Let R be a ring and let A be a subset of R. A map f : A→ R is commuting on A if [f(x),x]= 0 for all xeA where [x,y] = xy — yx. Suppose that R is a prime ring of characteristic ≠2 with extended centroid C. If L is a noncommutative Lie ideal of R and f:L→R an additive commuting map, then there is λe C and an additive map ∈: L→ C such that f(v) = λ(v)=λv+∈((v) for all veL.
On the iterates of derivations of prime rings
In this paper we study properties of associative derivations whose iterates are related in rather special ways to the original derivation, or to the iterates of another derivation. An associative derivation d: R -» R is an additive (or linear when appropriate) mapping on a ring R satisfying d(xy) — xd(y) 4- d(x)y for all x, y G R. A derivation d: R -* R is called inner if d(x) = (ad a)(x) for some a E JR where (ad a)(x) = [a, x] — ax — xa. In particular we ask when can the iterate of an inner derivation be an inner derivation? When can the iterates of two derivations commute? More precisely, we characterize elements α, b E R, R a prime ring, for which (ad a)"(x) = (ad b)(x) for all x E R, and we characterize derivations d: R -> R, 8: R -> R for which [dn(x), δn(y)] — 0 for all x, y E R, R prime. Applications are made to C*-algebras.