On derivations of standard operator algebras and semisimple H*-algebras

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) C L(X) be a standard operator algebra. Suppose we have a linear mapping D : A(X) - L(X) satisfying the relation D(A(3)) = D(A)A(2) + AD(A)A + A(2)D(A), for all A is an element of A(X). In this case D is of the form D(A) = AB - BA, for all A is an element of A(X) and some B is an element of L(X). We apply this result, which generalizes a classical result of Chernoff, to semisimple H*-algebras.