CHARACTERIZATIONS OF JORDAN LEFT DERIVATIONS ON SOME ALGEBRAS

A linear mapping delta from an algebra A into a left A-module M is called a Jordan left derivation if delta(A(2)) = 2A delta(A) for every A is an element of A. We prove that if an algebra A and a left A-module M satisfy one of the following conditions (1) A is a C*-algebra and M is a Banach left A-module; (2) A = Alg L with boolean AND{L_ : L is an element of J(L)} = (0) and M = B(X); and (3) A is a commutative subspace lattice algebra of a von Neumann algebra B and M = B (H) then every Jordan left derivation from A into M is zero. delta is called left derivable at G is an element of A if delta(AB) = A delta(B) + B delta(A) for each A, B is an element of A with AB = G. We show that if A is a factor von Neumann algebra, G is a left separating point of A or a nonzero self-adjoint element in A, and delta is left derivable at G, then delta 0.