Ortho-isomorphisms of von Neumann algebras
Minghui Ma, Weijuan Shi
TL;DR
The paper investigates maps between von Neumann algebras that preserve orthogonality in a strong sense via ortho-additivity and bijective ortho-preservation. It first shows that any ortho-additive range-contractive map on a algebra without $\mathrm{I}_1$ summands must act as left multiplication by $\\Phi(I)$, i.e., $\\Phi(A)=A\\Phi(I)$. Building on Dye’s projection-lattice framework, it then proves that any ortho-additive ortho-isomorphism decomposes as $\\varphi(A)=\\Phi(A)\\varphi(I)$ with $\\Phi$ the direct sum of a $*$-isomorphism and a conjugate $*$-isomorphism; the type $\mathrm{I}_2$ case is treated separately. The results extend Dye/Uhlhorn-type rigidity to broader von Neumann algebra settings and provide a complete structural description of orthogonality-preserving transformations.
Abstract
Suppose $\mathscr M$ and $\mathscr N$ are von Neumann algebras. Two operators $A$ and $B$ in $\mathscr M$ are said to be orthogonal if $A^*B=0$, meaning their ranges are orthogonal. Let $\varphi\colon\mathscr M\to\mathscr N$ be a map. We say that $\varphi$ is an ortho-isomorphism if it is bijective and satisfies that $A^*B=0$ if and only if $\varphi(A)^*\varphi(B)=0$ for all $A,B\in\mathscr M$. The map $\varphi$ is called ortho-additive if the additive relation $\varphi(A+B)=\varphi(A)+\varphi(B)$ holds for all $A,B\in \mathscr M$ with $A^*B=0$. 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.