Bilocal derivations of standard operator algebras

Abstract

In this paper, we shall show the following two results: (1) Let A be a standard operator algebra with I, if (D is a linear mapping on A which satisfies that 4>(T) maps ker T into ranT for all T E A, then (D is of the form P(T) = TA + BT for some A, B in B(X). (2) Let X be a Hilbert space, if b is a norm-continuous linear mapping on B(X) which satisfies that 4>(P) maps ker P into ran P for all self-adjoint projection P in then (D is of the form P(T) = TA + BT for some A, B in B(X). In what follows X stands for a Banach space (or Hilbert space) and X* for its norm dual. We denote by (x, f) the duality pairing between elements E X* and E X, and we use the symbols B(X), L(X), F(X), I and x 0 f to denote the set of all linear bounded operators on X, the set of all linear mappings on X, the set of all finite rank operators on X, the identity operator and the rank one operator (*, f)x on X, respectively. If A is a Banach algebra, and A1 is a Banach subalgebra of A, we say that a linear mapping 4D: A1 -A is a derivation if (ab) = (a)b + a4(b) for any a and b in A1. The derivation 1D is called inner if there exists an element a in A such that 4(b) = ba ab for any b in A1. We say that a linear mapping 4D: A1 -A is a local derivation if for every a in A1, there exists a derivation &a: A1 -A, depending on a, such that 4(a) = ba(a). A linear mapping 4i is called a Jordan derivation if (a 2) = a4(a) + a4(a) for every a in A1. We give the notion of bilocal derivation as follows: Definition 1. If A is a Banach subalgebra of a linear mapping pp: A B(X) is called a bilocal derivation if for every T in A and u in X, there exists a derivation 6T,u A -B(X), depending on T and u, such that 4D(T)u = 5T,U(T)U. Definition 2. Let X be a Banach space, a Banach subalgebra A of B(X) is called a standard operator algebra if A contains F(X). D. R. Larson and A. R. Sourour [5] have proved that every local derivation on B(X) is a derivation. R. Kadison [4] and M. Bresar [1] have discussed normcontinuous local derivations on von Neumann algebras. It is obvious that every Received by the editors June 14, 1995 and, in revised form, November 8, 1995. 1991 Mathematics Subject Classification. Primary 47D30, 47D25, 47B47.