A linear mapping from an algebra into a left -module is called a Jordan left derivation if for every . We prove that if an algebra and a left -module satisfy one of the following conditions—(1) is a -algebra and is a Banach left -module; (2) with and ; and (3) is a commutative subspace lattice algebra of a von Neumann algebra and —then every Jordan left derivation from into is zero. is called left derivable at if for each with . We show that if is a factor von Neumann algebra, is a left separating point of or a nonzero self-adjoint element in , and is left derivable at , then .
"Characterizations of Jordan left derivations on some algebras." Banach J. Math. Anal. 10 (3) 466 - 481, July 2016. https://doi.org/10.1215/17358787-3599675