An identity related to derivations of standard operator algebras and semisimple H∗ -algebras

Abstract

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 A(X) ⊂ L(X) be a standard operator algebra. Suppose D : A(X) → L(X) is a linear mapping satisfying the relation  for all A ∈ A(X). In this case D is of the form D(A) = AB − BA, for all A ∈ A(X) and some B ∈ L(X), 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 n ≥ 2, a ring R is said to be n−torsion free, if for x ∈ R, nx = 0 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 D(xy) = D(x)y + xD(y) 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 D(xy) = D(x)y + xD(y) holds for all pairs x,y ∈ R and is called a Jordan derivation in case D(x²) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = ax − xa 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 (x, yz^∗) = (xz, y) = (z, x^∗ y) 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.