## Banach Journal of Mathematical Analysis

### 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 $\delta(A^{2})=2A\delta(A)$ 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 $C^{*}$-algebra and $\mathcal{M}$ is a Banach left $\mathcal{A}$-module; (2) $\mathcal{A}=\operatorname {Alg}\mathcal{L}$ with $\cap\{L_{-}:L\in\mathcal{J}_{\mathcal{L}}\}=(0)$ and $\mathcal{M}=B(X)$; and (3) $\mathcal{A}$ is a commutative subspace lattice algebra of a von Neumann algebra $\mathcal{B}$ and $\mathcal{M}=B(\mathcal{H})$—then every Jordan left derivation from $\mathcal{A}$ into $\mathcal{M}$ is zero. $\delta$ is called left derivable at $G\in\mathcal{A}$ if $\delta(AB)=A\delta(B)+B\delta(A)$ 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 $\delta\equiv0$.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 466-481.

Dates
Accepted: 18 August 2015
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.bjma/1463153911

Digital Object Identifier
doi:10.1215/17358787-3599675

Mathematical Reviews number (MathSciNet)
MR3504180

Zentralblatt MATH identifier
1351.47027

#### Citation

An, Guangyu; Ding, Yana; Li, Jiankui. Characterizations of Jordan left derivations on some algebras. Banach J. Math. Anal. 10 (2016), no. 3, 466--481. doi:10.1215/17358787-3599675. https://projecteuclid.org/euclid.bjma/1463153911