Characterizations of Jordan left derivations on some algebras

Abstract

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