Ortho-isomorphisms of von Neumann algebras

Abstract

Suppose $\mathcal{M}$ and $\mathcal{N}$ are von Neumann algebras. Two operators $A$ and $B$ in $\mathcal{M}$ are said to be orthogonal if $A^{\ast }B=0$, meaning their ranges are orthogonal. Let $\phi :\mathcal{M}\to \mathcal{N}$ be a map. We say that $\phi$ is an ortho-isomorphism if it is bijective and satisfies that $A^{\ast }B=0$ if and only if $\phi (A{)}^{\ast }\phi (B)=0$ for all $A,B\in \mathcal{M}$. The map $\phi$ is called ortho-additive if the additive relation $\phi (A+B)=\phi (A)+\phi (B)$ holds for all $A,B\in \mathcal{M}$ with $A^{\ast }B=0$. In this paper, we characterize the complete structure of ortho-additive ortho-isomorphisms between von Neumann algebras, which is an analogue of Dye's theorem and Uhlhorn's theorem.