Characterizing Derivations on Von Neumann Algebras by Local Actions

Let M be any von Neumann algebra without central summands of type I-1 and P a core-free projection with the central carrier I. For any scalar xi, it is shown that every additive map L on M satisfies L (Lambda B-xi B Lambda) = L(Lambda)B - xi BL(Lambda) + Lambda L(B) - xi L(B)Lambda whenever AB = P if and only if (1) xi = 1, L = phi + h, where phi is an additive derivation and h is a central valued additive map vanishing on AB - BA with AB = P; ( 2) xi not equal 1, L is a derivation with L (xi A) = xi L(A) for each A is an element of M.