An equation on operator algebras and semisimple H*-algebras

Abstract

Abstract. In this paper we prove the following result: Let X be a Banach space over the real or complex field F and let L(X) be the algebra of all bounded linear operators on X. Suppose there exists an additive mapping T : A(X) → L(X), where A(X) ⊂ L(X) is a standard operator algebra. Suppose that T (A 3 ) = AT (A)A holds for all A ∈ A(X). In this case T is of the form T (A) = λA for any A ∈ A(X) and some λ ∈ F. This result is applied to semisimple H * −algebras.This research is related to the work of Molnár [8] and is a continuation of our work [9,10]. Throughout, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, if nx = 0, x ∈ R implies x = 0. The commutator xy − yx will be denoted by [x, y] . is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = [a, x] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [6] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of Herstein's result can be found in [3] . Cusack [5] generalized Herstein's result to 2−torsion free semiprime rings (see also [2] for an alternative proof).

Centralizer’s applications to the inverse along an element

“…In 1966, Kellogg [5] further investigated centralizers in H * -algebra. Later, Vukman [13,14] and Zalar [16] studied centralizers in operator algebras and semiprime rings, respectively.…”

Approximately n--Jordan derivations: A fixed point approach

“…Cusack [5] generalized Herstein's result to 2-torsion-free semiprime rings (see also [3] for an alternative proof). For some other results concerning derivations on prime and semiprime rings, Jordan derivations and n-Jordan derivations, we refer to [11,12,21,34,35].…”