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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. [ABSTRACT FROM AUTHOR]

Let \mathcal M be a semifinite factor with a faithful normal semifinite tracial weight \tau, and \mathscr P the set of all projections in \mathcal M. Denote by \mathscr P_{c} the Grassmann space of all projections in \mathscr P with trace c, where c is a positive real number. A map \psi : \mathscr P_c\rightarrow \mathscr P_c is called an ortho-isomorphism if \psi is a bijection of \mathscr P_c onto \mathscr P_c satisfying, for all P,Q\in \mathscr P_c, P\perp Q if and only if \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 \mathscr P_c in a semifinite factor. And we show that an ortho-isomorphism \psi : \mathscr P_c\rightarrow \mathscr P_c can be extended to a Jordan *-isomorphism \rho of \mathcal M onto \mathcal M. As an application, we obtain the structure of surjective isometries on \mathscr P_c with respect to strictly increasing unitarily invariant norms. [ABSTRACT FROM AUTHOR]