An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra1

摘要

In this paper we prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X, and let be a standard operator algebra. Suppose is a linear mapping satisfying the relation . In this case D is of the form and some , which means that D is a linear derivation. In particular, D is continuous. We apply this result, which generalizes a classical result of Chernoff, to semisimple H*- algebras. This research has been motivated by the work of Herstein [4], Chernoff [2] and Molnár [5] and is a continuation of our recent work [8] and [9] .Throughout, R will represent an associative ring. Given an integer , a ring R is said to be n?torsion free, if for implies x = 0. Recall that a ring R is prime if for a, b R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B A is called a linear derivation in case holds for all pairs x, y R. In case we have a ring R an additive mapping D : R R is called a derivation if holds for all pairs x, y R and is called a Jordan derivation in case is fulfilled for all x R. A derivation D is inner in case there exists a R, such that holds for all x R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [4] asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. Cusack [3] generalized Herstein?s result to 2 -torsion free semiprime rings. Let us recall that a semisimple H*-algebra is a semisimple Banach * -algebra whose norm is a Hilbert space norm such that is fulfilled for all x, y, z A (see [1]). Let X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra A(X) L(X) is said to be standard in case F(X) A(X). Let us point out that any standard algebra is prime, which is a consequence of Hahn-Banach theorem.