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
.