5.1 Multiple Systems
89
Hence for any nonzero projector s ≤ s ω one has ω(s) = 0 and π ω (s) = I ω := the
identity of L(H ω ). From this follows ω(s ω − s) = 0 and s ω − s = (s ω − s)(I −
s ω ) = 0, so that s ω is a minimal projector in M G . This proves that ω ∈ M ∗ .
5.1.18 For any state ω ∈ S g , the measures μ
ω
ξ (ξ ∈ g) are probability (σ-additive)
regular Borel measures on R, due to normality of ω ∈ S(M G ), (5.1.70). Define the
subset S
d
g ⊂ S(A
):
S
d
g := {ω ∈ S g : ω(X ξ ) is finite for all ξ ∈ g},
(5.1.75)
where ω(X ξ ) is defined in (5.1.72). Due to (5.1.61), the set S
d
g is σ
∗
G −invariant. For
any f ∈ L
1
(R, μ
ω
ξ ) define
ω( f (X ξ )) :=
R
f (λ) ω(E ξ (dλ)).
(5.1.76)
Any state ω ∈ S g which is mapped into M ∗ , e.g. any pure state ω ∈ S g , belongs
to S
d
g and, moreover,
ω(X
2
ξ ) = [ω(X ξ )]
2 for all ξ ∈ g.
(5.1.77)
Denote F ω (ξ) := ω(X ξ ) for ω ∈ S
d
g . The mapping
F : S
d
g → g
∗
, ω → F(ω) := F ω ; F ω (ξ) := ω(X ξ ), ξ ∈ g,
(5.1.78)
maps orbits of σ
∗
G in S
d
g onto orbits of Ad
∗
(G) in g
∗ . Let ω ∈ S
d
g and O ω := σ
∗
G ω
be the corresponding orbit. If (5.1.77) is valid for ω then it is valid for all the states
in O ω , as it is seen from (5.1.61). We shall call orbits O ω ⊂ S
d
g satisfying (5.1.77)
the G-macroscopically pure orbits, and similarly for single states; simply, we shall
use also (G-)pure orbits (resp. G-pure states). The
set of all G-pure states will be denoted by E g (⊂ S
d
g ) .
The state ω ∈ E g need not be a (pure) state in ES(A
) or in ES g . But the following
assertion is valid:
5.1.19 Proposition. For ω ∈ S
d
g and its canonical image ω ∈ S(M G ) the following
statements are equivalent:
(i) ω ∈ E g ; (ii) ω ∈ M ∗ .
Proof. The implication (ii) ⇒ (i) is clear. Let ω ∈ E g and let μ ω be the Baire measure
on M corresponding to ω ∈ S ∗ (M G ). We shall prove that μ ω is concentrated on a
one point set {ω} ⊂ M ∗ . Let B n ⊂ R, n ∈ Z + , be an increasing absorbing sequence
of Borel sets, i.e. for any bounded Borel B ⊂ R there is some n B ∈ Z + that for all
89
Hence for any nonzero projector s ≤ s ω one has ω(s) = 0 and π ω (s) = I ω := the
identity of L(H ω ). From this follows ω(s ω − s) = 0 and s ω − s = (s ω − s)(I −
s ω ) = 0, so that s ω is a minimal projector in M G . This proves that ω ∈ M ∗ .
5.1.18 For any state ω ∈ S g , the measures μ
ω
ξ (ξ ∈ g) are probability (σ-additive)
regular Borel measures on R, due to normality of ω ∈ S(M G ), (5.1.70). Define the
subset S
d
g ⊂ S(A
):
S
d
g := {ω ∈ S g : ω(X ξ ) is finite for all ξ ∈ g},
(5.1.75)
where ω(X ξ ) is defined in (5.1.72). Due to (5.1.61), the set S
d
g is σ
∗
G −invariant. For
any f ∈ L
1
(R, μ
ω
ξ ) define
ω( f (X ξ )) :=
R
f (λ) ω(E ξ (dλ)).
(5.1.76)
Any state ω ∈ S g which is mapped into M ∗ , e.g. any pure state ω ∈ S g , belongs
to S
d
g and, moreover,
ω(X
2
ξ ) = [ω(X ξ )]
2 for all ξ ∈ g.
(5.1.77)
Denote F ω (ξ) := ω(X ξ ) for ω ∈ S
d
g . The mapping
F : S
d
g → g
∗
, ω → F(ω) := F ω ; F ω (ξ) := ω(X ξ ), ξ ∈ g,
(5.1.78)
maps orbits of σ
∗
G in S
d
g onto orbits of Ad
∗
(G) in g
∗ . Let ω ∈ S
d
g and O ω := σ
∗
G ω
be the corresponding orbit. If (5.1.77) is valid for ω then it is valid for all the states
in O ω , as it is seen from (5.1.61). We shall call orbits O ω ⊂ S
d
g satisfying (5.1.77)
the G-macroscopically pure orbits, and similarly for single states; simply, we shall
use also (G-)pure orbits (resp. G-pure states). The
set of all G-pure states will be denoted by E g (⊂ S
d
g ) .
The state ω ∈ E g need not be a (pure) state in ES(A
) or in ES g . But the following
assertion is valid:
5.1.19 Proposition. For ω ∈ S
d
g and its canonical image ω ∈ S(M G ) the following
statements are equivalent:
(i) ω ∈ E g ; (ii) ω ∈ M ∗ .
Proof. The implication (ii) ⇒ (i) is clear. Let ω ∈ E g and let μ ω be the Baire measure
on M corresponding to ω ∈ S ∗ (M G ). We shall prove that μ ω is concentrated on a
one point set {ω} ⊂ M ∗ . Let B n ⊂ R, n ∈ Z + , be an increasing absorbing sequence
of Borel sets, i.e. for any bounded Borel B ⊂ R there is some n B ∈ Z + that for all