Ortho-isomorphisms of Grassmann spaces in semifinite factors

Abstract

Let M \mathcal M be a semifinite factor with a faithful normal semifinite tracial weight τ \tau , and P \mathscr P the set of all projections in M \mathcal M . Denote by P c \mathscr P_{c} the Grassmann space of all projections in P \mathscr P with trace c c , where c c is a positive real number. A map ψ : P c → P c \psi : \mathscr P_c\rightarrow \mathscr P_c is called an ortho-isomorphism if ψ \psi is a bijection of P c \mathscr P_c onto P c \mathscr P_c satisfying, for all P , Q ∈ P c P,Q\in \mathscr P_c , P ⊥ Q P\perp Q if and only if ψ ( P ) ⊥ ψ ( Q ) \psi (P)\perp \psi (Q) . The aim of this paper is to establish a version of Uhlhorn’s theorem in the setting of semifinite factors. We give a complete characterization of ortho-isomorphisms on Grassmann space P c \mathscr P_c in a semifinite factor. And we show that an ortho-isomorphism ψ : P c → P c \psi : \mathscr P_c\rightarrow \mathscr P_c can be extended to a Jordan ∗ * -isomorphism ρ \rho of M \mathcal M onto M \mathcal M . As an application, we obtain the structure of surjective isometries on P c \mathscr P_c with respect to strictly increasing unitarily invariant norms.