Some zero product preserving additive mappings of operator algebras.

Let M be a von Neumann algebra without direct commutative summands, and let A be an arbitrary subalgebra of LS (M) containing M, where LS (M) is the ∗-algebra of all locally measurable operators with respect to M. Suppose δ is an additive mapping from A to LS (M) that satisfies the condition δ(A)B ∗ + Aδ(B) + δ(B)A ∗ + Bδ(A) = 0 whenever AB = BA = 0. In this paper, we prove that there exists an element Y in LS (M) such that δ(X) = XY - YX∗, for every X in A. [ABSTRACT FROM AUTHOR]