Characterizations of centralizable mappings on algebras of locally measurable operators

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.