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.