Characterizations of Centralizable Mappings on Algebras of Locally Measurable Operators

Abstract

A linear mapping ϕ from an algebra ${\cal A}$ into its bimodule ${\cal M}$ is called a centralizable mapping at G ∈ ${\cal A}$ if ϕ(AB) = ϕ(A)B = Aϕ(B) for each A and B in ${\cal A}$ with AB = G. In this paper, we prove that if ${\cal M}$ is a von Neumann algebra without direct summands of type I₁ and type II, ${\cal A}$ is a *-subalgebra with ${\cal M}$ ⊆ ${\cal A}$ ⊆ LS (${\cal M}$) and G is a fixed element in ${\cal A}$, then every continuous (with respect to the local measure topology t(${\cal M}$)) centralizable mapping at G from ${\cal A}$ into ${\cal M}$ is a centralizer.