Let M \mathcal M be a semifinite factor with a fixed faithful normal semifinite tracial weight τ \tau such that τ ( I ) = ∞ \tau (I)=\infty . Denote by P ( M , τ ) \mathscr P(\mathcal M,\tau ) the set of all projections in M \mathcal M and P ∞ ( M , τ ) = { P ∈ P ( M , τ ) : τ ( P ) = τ ( I − P ) = ∞ } \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 P ∞ ( M , τ ) \mathscr P^{\infty }(\mathcal M,\tau ) . We prove that every such map on P ∞ ( M , τ ) \mathscr P^{\infty }(\mathcal M,\tau ) can be extended to a Jordan ∗ * -isomorphism ρ \rho of M \mathcal M onto M \mathcal M .