Let
V
\mathtt {V}
be a standard subspace in the complex Hilbert space
H
\mathcal {H}
and
G
G
be a finite dimensional Lie group of unitary and antiunitary operators on
H
\mathcal {H}
containing the modular group
(
Δ
V
i
t
)
t
∈
R
(\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}}
of
V
\mathtt {V}
and the corresponding modular conjugation
J
V
J_{\mathtt {V}}
. We study the semigroup
\[
S
V
=
{
g
∈
G
∩
U
(
H
)
:
g
V
⊆
V
}
S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\}
\]
and determine its Lie wedge
L
(
S
V
)
=
{
x
∈
g
:
exp
(
R
+
x
)
⊆
S
V
}
\operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\}
, i.e., the generators of its one-parameter subsemigroups in the Lie algebra
g
\mathfrak {g}
of
G
G
. The semigroup
S
V
S_{\mathtt {V}}
is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form
G
exp
(
i
C
)
G \exp (iC)
, where
C
⊆
g
C \subseteq \mathfrak {g}
is an
Ad
(
G
)
\operatorname {Ad}(G)
-invariant closed convex cone.
Our main results assert that the Lie wedge
L
(
S
V
)
\operatorname {\textbf {L}}(S_{\mathtt {V}})
spans a
3
3
-graded Lie subalgebra in which it can be described explicitly in terms of the involution
τ
\tau
of
g
\mathfrak {g}
induced by
J
V
J_{\mathtt {V}}
, the generator
h
∈
g
τ
h \in \mathfrak {g}^\tau
of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup
S
V
S_{\mathtt {V}}
itself.