Centralizing mappings on von Neumann algebras

Abstract

Let R R be a ring with center Z ( R ) Z(R) . A mapping F F of R R into itself is called centralizing if F ( x ) x − x F ( x ) ∈ Z ( R ) F(x)x - xF(x) \in Z(R) for all x ∈ R x \in R . The main result of this paper states that every additive centralizing mapping F F on a von Neumann algebra R R is of the form F ( x ) = c x + ζ ( x ) , x ∈ R F(x) = cx + \zeta (x),x \in R , where c ∈ Z ( R ) c \in Z(R) and ζ \zeta is an additive mapping from R R into Z ( R ) Z(R) . We also consider α \alpha -derivations and some related mappings, which are centralizing on rings and Banach algebras

Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras

“…As in the proof of Corollary 2, we find that is a centralizing linear derivation. Therefore Brešar's result [19] yields our claim. Theorem 6.…”

“…With the help of Mathieu and Murphy's result [18], we arrive at the conclusion. (1) and the second case of assumption (2) in Theorem 1 and suppose that : A → A is a continuous centralizing mapping subjected to (18) and (19). Then maps A into its radical rad (A).…”

“…Proof. We first consider = 1 in (18) and := ( , ) in (19). It follows from Theorem 1 that is a ring derivation.…”

“…In this paper, we will establish the stability of functional inequalities with ring derivations and will deal with the problem for the radical range of these functional inequalities on Banach algebra by considering the base of noncommutative versions for the result of Singer and Wermer: Mathieu and Murphy [18] verify that every continuous centralizing linear derivation on a Banach algebra maps into the radical and Brešar [19] proved that every centralizing linear derivation on a semiprime Banach algebra maps into the intersection of the center and the radical. Moreover, Chaudhry and Thaheem [20] showed that two ring derivations on semiprime ring with suitable property map into the center.…”

Equivalent characterization of derivations on operator algebras

“…One popular topic is derivable maps. We say an additive map δ : R → M is derivable at β if δ(x y) = δ(x)y + xδ(y) for any x, y ∈ R with x y = β (see [1][2][3][4] and references therein). It is obvious that an additive map is a derivation if and only if it is derivable at every point.…”