Characterizations of centralizable mappings on algebras of locally measurable operators

A linear mapping $\varphi$ from an algebra $\mathcal{A}$ into its bimodule $\mathcal{M}$ 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 $AB=G$. In this paper, we prove that if $\mathcal{M}$ is a von Neumann algebra without direct summands of type ${\mathrm{I}}_1$ and type $\mathrm{I}\mathrm{I}$, $\mathcal{A}$ is a $\ast$-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.