Orthogonality preserving maps on a Grassmann space in semifinite factors

Abstract:

Let $\mathcal M$ be a semifinite factor with a fixed faithful normal semifinite tracial weight $\tau$ such that $\tau (I)=\infty$. Denote by $\mathscr P(\mathcal M,\tau )$ the set of all projections in $\mathcal M$ and $\mathscr P^{\infty }(\mathcal M,\tau )=\{P\in \mathscr P(\mathcal M,\tau ): \tau (P)=\tau (I-P)=\infty \}$. In this paper, as a generalization of Uhlhorn’s theorem, we establish the general form of orthogonality preserving maps on the Grassmann space $\mathscr P^{\infty }(\mathcal M,\tau )$. We prove that every such map on $\mathscr P^{\infty }(\mathcal M,\tau )$ can be extended to a Jordan $*$-isomorphism $\rho$ of $\mathcal M$ onto $\mathcal M$.