Orthogonality preserving maps on a Grassmann space in semifinite factors

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