Characterizations of Jordan left derivations on some algebras

Abstract

A linear mapping δ from an algebra A into a left A-module M is called a Jordan left derivation if δ(A2)=2Aδ(A) for every A∈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=AlgL with ∩{L−:L∈JL}=(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. δ is called left derivable at G∈A if δ(AB)=Aδ(B)+Bδ(A) for each A,B∈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 δ is left derivable at G, then δ≡0.