Characterizations of centralizers and derivations on some algebras

TL;DR

This paper investigates the properties of centralizable and derivable mappings on certain algebras, establishing conditions under which these mappings are necessarily centralizers or derivations, especially in von Neumann and triangular algebras.

Contribution

It characterizes full-centralizable and full-derivable points in von Neumann and triangular algebras, providing new insights into their structure and the behavior of related mappings.

Findings

Every point in a von Neumann algebra or a triangular algebra is a full-centralizable point.

A point in a von Neumann algebra is a full-derivable point iff its central carrier is the unit.

The paper extends the understanding of mappings at specific points in operator algebras.

Abstract

A linear mapping $\phi$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B=A\phi(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$, and $\phi$ is called a derivable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B+A\phi(B)$ 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.