Characterizations of centralizers and derivations on some algebras

Abstract

A linear mapping $\varphi$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G\in \mathcal{A}$ if $\varphi (AB)=\varphi (A)B=A\varphi (B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$, and $\varphi$ is called a derivable mapping at $G\in \mathcal{A}$ if $\varphi (AB)=\varphi (A)B+A\varphi (B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$. A point $G$ in $\mathcal{A}$ is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at $G$ is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.

13 pages