Characterizations of centralizable mappings on algebras of locally measurable operators

TL;DR

This paper characterizes continuous centralizable mappings on certain algebras of locally measurable operators, showing they are essentially centralizers under specific conditions involving von Neumann algebras.

Contribution

It proves that continuous centralizable mappings on algebras of locally measurable operators are centralizers, extending understanding of their structure in von Neumann algebra contexts.

Findings

Continuous centralizable mappings are centralizers.

Results apply to von Neumann algebras without type I_1 and II summands.

Establishes conditions under which mappings are centralizers.

Abstract

A linear mapping $\phi$ from an algebra $\mathcal{A}$ into its bimodule $\mathcal M$ 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 $AB=G$. In this paper, we prove that if $\mathcal M$ is a von Neumann algebra without direct summands of type $\mathrm{I}_1$ and type $\mathrm{II}$, $\mathcal A$ is a $*$-subalgebra with $\mathcal M\subseteq\mathcal A\subseteq LS(\mathcal{M})$ and $G$ is a fixed element in $\mathcal A$, then every continuous (with respect to the local measure topology $t(\mathcal M)$) centralizable mapping at $G$ from $\mathcal A$ into $\mathcal M$ is a centralizer.